# Given matrix A and matrix B. Find (if possible) the matrices: (a) AB (b) BA. A=begin{bmatrix}1 & -1 &15&0&-23&-2&2end{bmatrix} , B=begin{bmatrix}1 & 1 &01&-4&53&-1&2end{bmatrix}

Question
Matrices
Given matrix A and matrix B. Find (if possible) the matrices: (a) AB (b) BA.
$$A=\begin{bmatrix}1 & -1 &1\\5&0&-2\\3&-2&2\end{bmatrix} , B=\begin{bmatrix}1 & 1 &0\\1&-4&5\\3&-1&2\end{bmatrix}$$

2021-03-10
Step 1
(a) The given matrices are $$A=\begin{bmatrix}1 & -1 &1\\5&0&-2\\3&-2&2\end{bmatrix} , B=\begin{bmatrix}1 & 1 &0\\1&-4&5\\3&-1&2\end{bmatrix}$$
Step 2
Find AB
$$AB=\begin{bmatrix}1 & -1 &1\\5&0&-2\\3&-2&2\end{bmatrix}\begin{bmatrix}1 & 1 &0\\1&-4&5\\3&-1&2\end{bmatrix}$$
$$=\begin{bmatrix}1-1+3 & 1+4-1 &0-5+2\\5+0-6&5+0+2&0+0-4\\3-2+6&3+8-2&0-10+4\end{bmatrix}$$
$$=\begin{bmatrix}3 & 4 &-3\\-1&7&-4\\7&9&-6\end{bmatrix}$$
Step 3
b) Find BA
$$BA=\begin{bmatrix}1 & 1 &0\\1&-4&5\\3&-1&2\end{bmatrix}\begin{bmatrix}1 & -1 &1\\5&0&-2\\3&-2&2\end{bmatrix}$$
$$=\begin{bmatrix}1+5+0 & -1+0+0 &1-2+0\\1-20+15&-1+0-10&1+8+10\\3-5+6&-3+0-4&3+2+4\end{bmatrix}$$
$$=\begin{bmatrix}6 & -1 &-1\\-4&-11&19\\4&-7&9\end{bmatrix}$$

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