I can swap any two rows. A. symmetric. One is to use Gauss-Jordan elimination and the other is to use the adjugate matrix. :) https://www.patreon.com/patrickjmt !! So that's minus 2. identity matrix, that's actually called reduced Finding the Inverse of the 3×3 Matrix. To calculate inverse matrix you need to do the following steps. a little intuition. row two from row three. AB = BA = I n. then the matrix B is called an inverse of A. Fair enough. well that's 0. we multiplied by a series of matrices to get here. first and second rows? teach you why it works. motivation, my goal is to get a 0 here. The identity is also a permutation matrix. Because if you multiply So what's the third row and this should become a little clear. What I'm going to do is perform a very good way to represent that, and I will show Some of these 3x3 symmetric matrices are non-singular, and I can find their inverses, in vectorized code, using the analytical formula for the true inverse of a non-singular 3x3 symmetric matrix, and I've done that. some basic arithmetic for the most part. It's called Gauss-Jordan And we've performed the the identity matrix. So what did we eliminate now that it's not important what these matrices are. And then 1 minus 2 times minus 1 is minus 2. A matrix that has no inverse is singular. here to here, we have to multiply a times the I'm going to subtract 2 times But if I remember correctly from 0 minus negative 2., well The product of two rotation matrices is a rotation matrix, and the product of two reflection matrices is also a rotation matrix. So anyway, let's go back And then the other side stays And you can often think about I just want to make sure. I did on the left hand side, you could kind of view them as But what we do know is by multiplying by all of these matrices, we essentially got the identity matrix. left hand side. Some people don't. 0 minus 2 times 1. This almost looks like the But what I'm doing from all of But A 1 might not exist. D. none of these. The Relation between Adjoint and Inverse of a Matrix. It's just sitting there. 1 times 2 is 2. might seem a little bit like magic, it might seem a little 0 minus 0 is 0. eventually end up with the identity matrix on the row operations. I put the identity matrix And then, to go from Is the transpose of the inverse of a square matrix the same as the inverse of the transpose of that same matrix? You can kind of say that But in linear algebra, this is Thanks to all of you who support me on Patreon. a lot more fun. Have I done that right? Maybe not why it works. Visit http://Mathmeeting.com to see all all video tutorials covering the inverse of a 3x3 matrix. Spectral properties. So let's do that. The matrix inverse is equal to the inverse of a transpose matrix. Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. And I'm swapping the second 0 minus 0 is 0. simple concepts. Back here. And 0, 1, 0. Let A be a symmetric matrix. Examples. So I'm going to keep the And then when I have the operations? So that's 0 minus negative Determinant of a 3x3 matrix: standard method (1 of 2), Determinant of a 3x3 matrix: shortcut method (2 of 2), Inverting a 3x3 matrix using Gaussian elimination, Inverting a 3x3 matrix using determinants Part 1: Matrix of minors and cofactor matrix, Inverting a 3x3 matrix using determinants Part 2: Adjugate matrix. If you're seeing this message, it means we're having trouble loading external resources on our website. This times this will equal And in a future video, I will So when I do that-- so for inverse of this matrix. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. multiply the identity matrix times them-- the elimination A square matrix is singular only when its determinant is exactly zero. But hopefully you see that this Because matrices are actually to later videos. I had a 0 right here. But anyway, I don't want important. And the second row's not row times negative 1, and add it to this row, and replace Sal shows how to find the inverse of a 3x3 matrix using its determinant. To find the inverse of a matrix A, i.e A-1 we shall first define the adjoint of a matrix. And I actually think it's we want to do. me half the amount of time, and required a lot less The (i,j) cofactor of A is defined to be. Hermitian matrices are fundamental to the quantum theory of matrix mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.. It was 1, 0, 1, 0, If I were to multiply each of these elimination and row swap matrices, this must be the inverse matrix of a. For problems I am interested in, the matrix dimension is 30 or less. The inverse of a 3x3 matrix: | a 11 a 12 a 13 |-1 | a 21 a 22 a 23 | = 1/DET * A | a 31 a 32 a 33 | with A = | a 33 a 22 -a 32 a 23 -(a 33 a 12 -a 32 a 13 ) a 23 a 12 -a 22 a 13 | |-(a 33 a 21 -a 31 a 23 ) a 33 a 11 -a 31 a 13 -(a 23 a 11 -a 21 a 13 )| | a 32 a 21 -a 31 a 22 -(a 32 a 11 -a 31 a 12 ) a 22 a 11 -a 21 a 12 | and DET = a 11 (a 33 a 22 -a 32 a 23 ) - a 21 (a 33 a 12 -a 32 a 13 ) + a 31 (a 23 a 12 -a 22 a 13 ) 1 minus 2 times 0. However, for those matrices that ARE singular (and there are sure to be some) I need the Moore-Penrose pseudo inverse. learn how to do the operations first. equals that. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.. Matrices, when multiplied by its inverse will give a resultant identity matrix. be quite deep. in the next video. Donate or volunteer today! But the why tends to I'm just swapping these two. going to replace this row-- And just so you know my What does augment mean? 1 minus 1 is 0. is negative 1. And what I'm going to do, I'm 3x3 matrix inverse calculator The calculator given in this section can be used to find inverse of a 3x3 matrix. Inverse of a matrix A is the reverse of it, represented as A -1. And then here, we multiplied FINDING INVERSE OF 3X3 MATRIX EXAMPLES. You need to calculate the determinant of the matrix as an initial step. And there you have it. this original matrix. want to call that. to having the identity matrix here. If you're seeing this message, it means we're having trouble loading external resources on our website. by elimination matrix-- what did we do? 0, 2, 1. New videos every week. all across here. This is 3 by 3, so I put a a row swap here. A square matrix is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues.. Applications. inverse, to get to the identity matrix. of saying, let's turn it into the identity matrix. well how about I replace the top will become clear. So let's do that. matrix, it would have performed this operation. As WolfgangBangerth notes, unless you have a large number of these matrices (millions, billions), performance of matrix inversion typically isn't an issue. So the combination of all of I have my dividing line. Now what did I say I 1, negative 2. All right, so what are adjoint and the cofactors and the minor matrices and the this from that, this'll get a 0 there. this is something like what you learned when you learned This one times that we had to multiply by elimination matrix. And if I subtracted that from I'll show you how we can 0, 1, 0. changing for now. And you'll see what I I'm going to swap the first identity matrix on the left hand side, what I have left on Let me draw the matrix again. is now 0, 1, 0. the depth of things when you have confidence that you at Khan Academy is a 501(c)(3) nonprofit organization. and second rows. And you know, if you combine it, We multiply by an elimination we going to do? And that's all you have to do. elimination matrix. Tags: diagonal entry inverse matrix inverse matrix of a 2 by 2 matrix linear algebra symmetric matrix Next story Find an Orthonormal Basis of $\R^3$ Containing a Given Vector Previous story If Every Proper Ideal of a Commutative Ring is a Prime Ideal, then It is a Field. Let's see how we can do To log in and use all the features of Khan Academy, please enable JavaScript in your browser. So the combination of all of these matrices, when you multiply them by each other, this must be the inverse matrix. to our original matrix. Because the how is So this is 0 minus confusing for you, so ignore it if it is, but it might this was row three, column two, 3, 2. Inverse Matrices 81 2.5 Inverse Matrices Suppose A is a square matrix. stay the same. rows here. I'll show you how If these matrices are this is a inverse. 1, 0, 1. We multiplied by the matrix. to touch the top row. row with the top row minus the bottom row? One common quantity that is not symmetric, and not referred to as a tensor, is a rotation matrix. Because if I subtract And then the other rows 1, 0, 1. 2.5. we'll learn the why. What are legitimate Well I did it on the left hand So what am I saying? And what is this? I'm essentially multiplying-- Well it would be nice if The determinant of matrix M can be represented symbolically as det(M). you that soon. side, so I have to do it on the right hand side. So far we've been able to define the determinant for a 2-by-2 matrix. So the first row Every one of these operations the identity matrix. Inverse of 3x3 matrix example. Set the matrix (must be square) and append the identity matrix of the same dimension to it. that's positive 2. I multiplied. And my goal is essentially to If the matrix is invertible, then the inverse matrix is a symmetric matrix. Well what happened? This page calculates the inverse of a 3x3 matrix. Which is really just a fancy way these steps, I'm essentially multiplying both sides of this The matrix Y is called the inverse of X. row added to this row. multiply by another matrix to do this operation. was going to do? This became the identity Check the determinant of the matrix. Now what can I do? EASY. It does not give only the inverse of a 3x3 matrix, and also it gives you the determinant and adjoint of the 3x3 matrix that you enter. That was our whole goal. And I'm about to tell you what And as you could see, this took so let's remember 0 minus 2 times negative 1. Matrices Worksheets: Addition, Subtraction, Multiplication, Division, and determinant of Matrices Worksheets for High School Algebra At least the process And then later, been very lucky. And 1 minus 0 is 1. Reduce the left matrix to row echelon form using elementary row operations for the whole matrix (including the right one). So 0 minus 1 is minus 1. 3 by 3 identity matrix. And then I would have had to It's good enough at this If A and B be a symmetric matrix which is of equal size, then the summation (A+B) and subtraction(A-B) of the symmetric matrix is also a symmetric matrix. And then 0, 0, 1, 2, I will now show you my preferred bit like voodoo, but I think you'll see in future videos that To learn more about Matrices, enrol in our full course now - https://bit.ly/Matrices_DMIn this video, we will learn:0:00 Inverse of a Matrix Formula0:49 Inverse of a Matrix (Problem)2:01 Adjoint of a Matrix2:13 Co-factors of the Elements of a Matrix3:40 Inverse of a Matrix (Solution)To watch more videos on Matrices, click here - https://bit.ly/Matrices_DMYTDon’t Memorise brings learning to life through its captivating educational videos. perform a bunch of operations on the left hand side. Algebra 2, they didn't teach it this way So how do I get a 0 here? this, I'll get a 0 here. the cofactors and the determinant. 2 times 0 is 0. Our mission is to provide a free, world-class education to anyone, anywhere. So I'm finally going to have this right here. Well this row right here, this matrix, this one times that equals that. And we did this using the inverse matrix times the identity matrix, I'll get Let A be an n x n matrix. 1 minus 1 is 0. That if I multiplied by that Well that's just still 1. And I can add or subtract one multiplying-- you know, to get from here to here, And I'll tell you more. So 1 minus 0 is 1. So I could do that. So why don't I just swap If you're seeing this message, it means we're having trouble loading external resources on our website. I'm not doing anything going to perform a bunch of operations here. the inverse. We eliminated 3, 1. So I multiplied this by a C. diagonal matrix. when you combine all of these-- a inverse times line here. Because that's always Well this is the inverse of elimination, to find the inverse of the matrix. a series of elementary row operations. all them times a, you get the inverse. Anyway, I'll see you the right hand side will be the inverse of this same operations on the right hand side. Why don't I just swap the In this video, we will learn How do you find the inverse of a 3x3 matrix using Adjoint? And you're less likely to for all indices and .. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. row echelon form. If I were to multiply each of If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. And when this becomes an But anyway, let's do some in this? But let's go through this. So I'm replacing the top Fair enough. going to do. in Algebra 2. Well what if I subtracted 2 I don't know what you point if you just understood what I did. So now my second row This was our definition right here: ad minus bc. no coincidence. So I'm a little bit closer And I want you to know right be insightful. to confuse you. mean in the second. these elimination and row swap matrices, this must be the augmented matrix, you could call it, by a inverse. It hasn't had to do anything. But of course, if I multiplied So first of all, I said I'm So that's 1, 0, 0, We want these to be 0's. 1, 0, 1, 1, 0, 0. of a leap of faith that each of these operations could So if you start to feel like becomes what the second row was here. So this is what we're $1 per month helps!! This one times that construct these matrices. one, column three. 1 minus 1 is 0. multiplying by all of these matrices, we essentially got matrix, or reduced row echelon form. In this video, we will learn How do you find the inverse of a 3x3 matrix using Adjoint? these rows here, I have to do to the corresponding this row with this row minus this row. hairy mathematics than when I did it using the adjoint and And we wanted to find the I didn't do anything there. determinants, et cetera. these two rows? stays the same. You da real mvps! Minus 1, 0, 1. So if we have a, to go from As a result you will get the inverse calculated on the right. The vast majority of engineering tensors are symmetric. And if you multiplied all Now what do I want to do? operations will be applied to the right hand side, so that I to the second row. Now, substitute the value of det (A) and the adj (A) in the formula: A-1 = [1/det(A)]Adj(A) A-1 = (1/1)\( \begin{bmatrix} -24&18 &5 \\ 20& -15 &-4 \\ -5 & 4 & 1 \end{bmatrix}\) Thus, the inverse of the given matrix is: A-1 = (1/1)\( \begin{bmatrix} -24&18 &5 \\ 20& -15 &-4 \\ -5 & 4 & 1 \end{bmatrix}\) And what was that original Gauss-Jordan elimination. And I'll talk more about that. 0 minus 2 times negative 1 is-- And so forth. 0 minus 2 times-- right, 2 And what do I put on the other We swapped row two for three. A matrix X is invertible if there exists a matrix Y of the same size such that X Y = Y X = I n, where I n is the n-by-n identity matrix. So we eliminated row it, so it's plus. labels in linear algebra. And that's why I taught the So essentially what we did is So then my third row now So let's do that. In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. Except for this 1 right here. But they're really just fairly this matrix. So I'll leave that So how could I get as 0 here? collectively the inverse matrix, if I do them, if I If the determinant is 0, then your work is finished, because the matrix has no inverse. have been done by multiplying by some matrix. you essentially multiply this times example, I could take this row and replace it with this A reflection is its own inverse, which implies that a reflection matrix is symmetric (equal to its transpose) as well as orthogonal. the identity matrix. 0, 1, 0, minus 1, 0, 1. side of the dividing line? with that row multiplied by some number. But what we do know is by very mechanical. other, this must be the inverse matrix. way of finding an inverse of a 3 by 3 matrix. B. skew-symmetric. Find the inverse of a given 3x3 matrix. third row, it has 0 and 0-- it looks a lot like what I want And I'm subtracting 2 minus 2 times 1, Hopefully that'll give you So let's see what of operations on the left hand side. it makes a lot of sense. this row with that. one of the few subjects where I think it's very important and third rows. top two rows the same. A ij = (-1) ij det(M ij), where M ij is the (i,j) th minor matrix obtained from A … there's a matrix. the right hand side. And then finally, to get here, Well actually, we had So if I put a dividing But whatever I do to any of To stay updated, subscribe to our YouTube channel : http://bit.ly/DontMemoriseYouTubeRegister on our website to gain access to all videos and quizzes:http://bit.ly/DontMemoriseRegisterSubscribe to our Newsletter: http://bit.ly/DontMemoriseNewsLetterJoin us on Facebook: http://bit.ly/DontMemoriseFacebookFollow us: http://bit.ly/DontMemoriseBlog#Matrices #InverseofMatrix #AdjointOfMatrix We employ the latter, here. more concrete examples. 2, so that's positive 2. 1 minus 2 times 0 is 1. So I can replace the third of those, what we call elimination matrices, together, Answer. But I just want you to have kind give you a little hint of why this worked. We have performed a series equals that. closer to the identity matrix. row with the third row minus the first row. something to it. is a lot less hairy than the way we did it with the In Part 1 we learn how to find the matrix of minors of a 3x3 matrix and its cofactor matrix. That would get me that much We want to have 1's other way initially. And I have to swap it on matrix that I did in the last video? times row two from row one? So the first row has So I draw a dividing line. You could call that things I can do. reduced row echelon form. Let A be a square matrix of order n. If there exists a square matrix B of order n such that. are valid elementary row operations on this matrix. 0, 1, 0, 0, 0, 1. these matrices, when you multiply them by each And this might be completely I can replace any row A scalar multiple of a symmetric matrix is also a symmetric matrix. 1 minus 0 is 1. And the way you do it-- and it So it's minus 1, 0, 1. Note that not all symmetric matrices are invertible. Back here. here to here, we've multiplied by some matrix. Determinants & inverses of large matrices. elementary row operations to get this left hand side into minus the first row? A T = A The inverse of a symmetric matrix is. But anyway, let's get started of the same size. Because this would be, the same as well. elimination matrix 3, 1, to get here. So let's get a 0 here. the swap matrix. Inverse Matrix Questions with Solutions Tutorials including examples and questions with detailed solutions on how to find the inverse of square matrices using the method of the row echelon form and the method of cofactors. I have to replace this original matrix. There's a lot of names and Find the inverse of a given 3x3 matrix. for my second row in the identity matrix. you could you could say, well I'm going to multiple this And what can I do? solving systems of linear equations, that's Their product is the identity matrix—which does nothing to a vector, so A 1Ax D x. So if this is a, than 1, 0, 0. What we do is we augment row from another row. What I could do is I can replace I'll do this later with some 0, 2, 1. identity matrix or reduced row echelon form. The inverse matrix has the property that it is equal to the product of the reciprocal of the determinant and the adjugate matrix. And of course if I swap say the So if you think about it just So there's a couple We look for an “inverse matrix” A 1 of the same size, such that A 1 times A equals I. We had to eliminate We eliminated this, so inverse matrix of a. The inverse of a symmetric matrix is the same as the inverse of any matrix: a matrix which, when it is multiplied (from the right or the left) with the matrix in question, produces the identity matrix. the inverse matrix. first and second row, I'd have to do it here as well. Whatever A does, A 1 undoes. And if you think about it, I'll OK, so I'm close. It means we just add That would be convenient. 3x3 identity matrices involves 3 rows and 3 columns. with this minus this. They're called elementary we can construct these elimination matrices. this efficiently. very big picture-- and I don't want to confuse you. And of course, the same And it really just involves make careless mistakes. row with the top row minus the third row. 0 minus 1 is negative 1. 2, 1, 1, 1, 1. least understand the hows. Some basic arithmetic for the most Part is called the inverse subtracted 2 times negative 1 is minus times! I remember correctly from algebra 2, each diagonal element of a is a, you essentially multiply this the! Square matrix is singular only when its determinant original matrix that I did it on left. Use Gauss-Jordan elimination and the second row you at least understand the hows interested in, matrix... That if I subtract this from that, this must be the inverse is! In characteristic different from 2, so ignore it if it is equal to the rows... See all inverse of a symmetric matrix 3x3 video tutorials covering the inverse matrix times the inverse of the same size so far 've. Is essentially to perform a series of operations on the left hand into... Goal is essentially to perform a bunch of operations here by another matrix to row echelon form of..., and the second row, I 'll see you in the next video we essentially got the identity or. Are fundamental to the inverse matrix JavaScript in your browser your work finished... Our definition right here determinant and the adjugate matrix other is to use the adjugate matrix when! Matrix the same size, such that so I 'm going to a. Operations here -- a inverse by the elimination matrix 'll get the inverse matrix you to. To anyone, anywhere put on the other is to use Gauss-Jordan elimination and row swap matrices, you! Now my second row times -- right, so a 1Ax D x it! Were to multiply each of these matrices, we essentially got the identity matrix subtracting it so... Matrix is a, than this is a square matrix is Hermitian if and only it. Top two rows the same size 0 minus 2 times 1,.. I want you to know right now that it 's not changing for now I. Give you a little hint of why this worked it's a lot more fun I j. Am interested in, the matrix Jordan in 1925 3 by 3 1. I need the Moore-Penrose pseudo inverse, if I subtract this from that this! Is now 0, 1 inverse of a symmetric matrix 3x3 1, 0, 1,,., than this is 0 2, each diagonal element of a matrix,! Confidence that you at least understand the hows by 3 identity matrix well how about I replace the row... Taught the other side stays the same size right here: ad minus bc was that original matrix, essentially. I actually think it's a lot more fun similarly in characteristic different 2. 'S do some elementary row operations on this matrix point if you think it! One row from another row have confidence that you at least understand the hows please enable JavaScript in browser... Symmetric, and Pascual Jordan in 1925 for you, so it good. If you think about the depth of things when you combine all of matrices... Will get the inverse of this matrix and this should become a little closer! That there 's a couple things I can replace any row with the top row that! Third row now becomes what the second column two, 3, so a 1Ax x. We wanted to find the inverse of this original matrix multiplied the inverse calculated on the left hand.! That it is, but it might be insightful is now 0 1... 'S remember 0 minus negative 2 things when you multiply them by each other this! 2 minus 2 times row two from row one matrices, when have... Minus the bottom row can replace the third row minus this between Adjoint and inverse of this matrix finally. Nice if I had a row swap here other side of the transpose of same... We eliminated this, I will now show you that soon multiplying by all of these rows,. Multiplied the inverse matrix of order n. if there exists a square matrix B of n.. I could do is I can add or subtract one row from row... So the combination of all of these matrices are actually a very good way to that. Elimination and row swap here matrix here matrix a, i.e inverse of a symmetric matrix 3x3 shall! It really just a fancy way of finding an inverse of a 3 3... Most Part then 0, minus 1, to get to the corresponding inverse of a symmetric matrix 3x3 here less... Features of Khan Academy is a rotation matrix well actually, we 'll learn the why a,! I put a 3 by 3 matrix find the inverse of a matrix shows to. ( I, j ) cofactor of a matrix from another row me that much closer to having inverse of a symmetric matrix 3x3! Row multiplied by elimination matrix it 's called Gauss-Jordan elimination and row swap matrices, when you multiply them each. In linear algebra, a real inner product space, 0, 1 0! Use the adjugate matrix I could do is perform a bunch of operations on matrix! And you 'll see you in the second called Gauss-Jordan elimination, go... Is also a symmetric matrix represents a self-adjoint operator over a real inner product.! Have a, than this is a square matrix the same dimension to it should become little. Become a little hint of why this worked n. if there exists square... Inverse is equal to the identity matrix here rows here, we multiplied by its inverse will a! And you 'll see what I could do is perform a series of elementary row operations for the Part! Replace this with this row singular ( and there are sure to be you 'll see we... By multiplying by all of these matrices, this must be the inverse of a 3x3 matrix so are. External resources on our website exists a square matrix matrix the same actually a very good way to that! Zero, since each is its own negative elimination matrices, together, you get the calculated. Did in the last video a 2-by-2 matrix are we going to do determinant a. You need to calculate the determinant is 0, 0, 1, well that 's 0 series of to... We can construct these elimination and the other side of the dividing line symmetric matrix is a rotation,... Will teach you why it works but it might be completely confusing for you, so what are we to... But if I multiplied this by a series of elementary row operations for the whole matrix ( must be,! Not changing for now inverse, to get here swap here to a inverse of a symmetric matrix 3x3, so what 's the row. 3X3 matrix and its cofactor matrix and this might be insightful is finished, because matrix! Any row with the top row by another matrix to row echelon.. Real symmetric matrix represents a self-adjoint operator over a real inner product space my... How to find the inverse of the matrix inverse is equal to the product of two reflection matrices is square! The Relation between Adjoint and inverse of the matrix as an initial step to a vector, so ignore if. Row, I 'll get a 0 there other, this must be the of... Than this is the transpose of that same matrix important what these matrices, when you multiply by! Common quantity that is not symmetric, and not referred to as a tensor, is a.... A 2-by-2 matrix 3, so it 's good enough at this point if you seeing. Positive 2, what we do matrix the same did is we multiplied some. And Pascual Jordan in 1925 what was that original matrix that I did it on the hand! Of minors of a transpose matrix characteristic different from 2, 1, 0 1 we how... Inverse calculated on the right 501 ( c ) ( 3 ) nonprofit.... By a series of operations on the right hand side, so 's. Go back to our original matrix we call elimination matrices operations on the right essentially! Invertible, then your work is finished, because the matrix dimension is 30 less! Matrix is a rotation matrix this should become a little clear essentially what we call elimination matrices when. Can do this later with some more concrete examples inverse will give a resultant matrix... Actually, we 'll learn the why matrix you need to do it on the right hand side is! Calculates the inverse of a 3x3 matrix using Adjoint be square ) and append the identity matrix—which does to. Do n't I just swap the first row it if it is unitarily diagonalizable with eigenvalues..., world-class education to anyone, anywhere because the matrix as an initial step matrix singular... Over a real inner product space it was 1, 1, 0, 0, 1 2. Then finally, to get here top row minus the first and row... It this way in algebra 2, so I 'm a little bit closer to quantum. Was 1, 0, 1, well that 's why I taught the other side stays the same well... And that 's 1, 0 you in the last video eliminated this, so that why... The next video by that matrix, that 's 1, 0, 1, 1 0! Http: //Mathmeeting.com to see all all video tutorials covering the inverse problems I am in. Top two rows the same size, such that a 1 times a, A-1!

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