# How do I invert this 3×3 matrix?

I've been following Khan Academy to try and teach myself how to invert matrices, howevever I tried to invert a 3×3 matrix and I am not sure where I have gone wrong.

Picture of my answer:

I tried to invert $A$ but my answer is wrong. I would appreciate any help given.

Link of KhanAcademy tutorial I was following: https://www.khanacademy.org/math/algebra-home/alg-matrices/alg-determinants-and-inverses-of-large-matrices/v/inverting-3x3-part-1-calculating-matrix-of-minors-and-cofactor-matrix

• Your original matrix has "2" but you use "-2" in calculating the determinant! Your original matrix has "1" but you use "10" in calculating the determinant! The determinant of the matrix you show is 10, not -100. – user247327 Feb 22 '17 at 17:14
• Your mistake: You need to calculate the determinant of $A$ which is $10$. – zoli Feb 22 '17 at 17:15
• You need the determinant of the original matrix, not of the cofactor matrix. – amd Feb 22 '17 at 22:00

## 2 Answers

Note that $A$ is diagonal. Convince yourself that merely taking reciprocals of the diagonal entries yields the inverse: $$A^{-1}=\begin{bmatrix} \frac12&0&0\\ 0&\frac15&0\\ 0&0&1\end{bmatrix}$$

• So all I actually had to do was get the reciporcals of those numbers? Was all the work i did irrelevant or did I just go wrong somewhere. Thanks – Alan Piggott Feb 22 '17 at 17:15
• @AlanPiggott Well if the matrix was not diagonal you would have to go through the process you have showed. As pointed out in the comments, you made your error in the determinant; it is 10 and not 100, and can be calculated with Sarrus's rule. As well as that, the middle entry in the cofactor should have been 2 and not -2, as pointed out in the comments. Those aside, your working is correct. – Parcly Taxel Feb 22 '17 at 17:18
• Ok I understand that thanks. I am still confused however on why the determinant is 10 and not 100. At the top right of my page that I have labeled ' matrix of numbers', I got 2x5 in the bottom right. Then I multplied 5x2x10. Where did I go wrong here? Thank you. – Alan Piggott Feb 22 '17 at 17:23
• @AlanPiggott It's not the determinant of $C$ you should take, it's the determinant of the original $A$. – Parcly Taxel Feb 22 '17 at 17:24
• @AlanPiggott Yes. I also need to mention that diagonal matrices are just lovely to work with. – Parcly Taxel Feb 22 '17 at 17:28

Just another note, it seems like you multiplied the central element of your matrix by $-1$ in the second line, which you're not meant to do - the diagonal elements are multiplied by $+1$ according to the Khan Academy tutorial.