# Thread: Finding inverse of a 3X3 Matrix

1. ## Finding inverse of a 3X3 Matrix

I've found inverses for 2X2 matrices in the past but have never done a 3X3. I looked at a few sites and they performed things such as row operations which I've never seen before. I tried to use the method myself but got more confused . Does anyone know a way that is easy to understand?

2. Have you learned how to reduce a matrix to row echelon form before? If so, then take your matrix and reduce it to row echelon form. For each step you do, you do it to the identity matrix. What is left after row reducing is your inverse matrix.

For example, say that your first step to reducing B to the identity matrix is dividing the entire first row by 4, then you do the same step to the identity matrix:

$\displaystyle \left[ \begin{array}{ccc} 4 & 5 & 8 \\ 7 & 6 & -1 \\ 0 & 9 & 2 \end{array} \Bigg| \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \: \: \Rightarrow$$\displaystyle {\color{white}.} \: \: \left[ \begin{array}{ccc} 1 & \frac{5}{4} & 2 \\ 7 & 6 & -1 \\ 0 & 9 & 2 \end{array} \Bigg| \begin{array}{ccc} \frac{1}{4} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$

See how what I did to the left side, I did to the right side? Now continue to row reduce the left matrix to the identity matrix and perform the same steps to the right matrix. Again, whatever your result is on the right side after row reducing the left matrix is your inverse.

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If not, you can always use this ghastly formula: $\displaystyle B^{-1} = \frac{adj(B)}{det(B)}$

$\displaystyle adj(B)$ is found by finding the cofactors of each entry of B and then taking its transpose.

Here's a resource that you may find helpful: Paul's Online Notes: Linear Algebra

3. oo I see what you did, but how did you know to divide by 4? Is there some way of telling what to multiply or divide by? Are we able to subtract and add too?