Q:

If A=[tex]\left[\begin{array}{ccc}-4&-3&5\\-5&-4&-2\\-1&-2&-4\end{array}\right][/tex] and B=[tex]\left[\begin{array}{ccc}3&-4&-1\\-5&-5&1\\-1&-3&2\end{array}\right][/tex], find AB.

Accepted Solution

A:
It's not C because[tex](AB)_{1,1}=A_{1,c}B_{r,1}=\begin{bmatrix}-4&-3&5\end{bmatrix}\begin{bmatrix}3\\-5\\-1\end{bmatrix}=-2[/tex]where [tex](AB)_{1,1}[/tex] denotes the element of [tex]AB[/tex] in row 1, column 1, [tex]A_{1,c}[/tex] denotes the first row of [tex]A[/tex], and [tex]B_{r,1}[/tex] denotes the first column of [tex]B[/tex].It's not A because[tex](AB)_{1,2}=A_{1,c}B_{r,2}=\begin{bmatrix}-4&-3&5\end{bmatrix}\begin{bmatrix}-4\\-5\\-3\end{bmatrix}=16[/tex]It's not D because[tex](AB)_{2,2}=A_{2,c}B_{r,2}=\begin{bmatrix}-5&-4&-2\end{bmatrix}\begin{bmatrix}-4\\-5\\-3\end{bmatrix}=46[/tex]So B must be the correct answer (and it is).