## College Algebra (6th Edition)

$\left[\begin{array}{ll} 16 & -16\\ -12 & 12\\ 0 & 0 \end{array}\right]$
A product of two matrices exists if the first has as many columns as the second matrix has rows. The product of an $m\times\underline{n}$ matrix $A$ and an $\underline{n}\times p$ matrix $B$ is an $m\times p$ matrix $AB$. The product BC exists because B is a 2$\times\underline{2}$ matrix, and C is a $\underline{2}\times$2 matrix. (BC) is 2$\times$2. The product of A ( a $3\times\underline{2}$ matrix) and BC ( a $\underline{2}\times$2 matrix) exists and is a $3\times 2$ matrix. $BC=\left[\begin{array}{ll} 5(1)+1(-1) & 5(-1)+1(1)\\ -2(1)-2(-1) & -2(1)-2(-1) \end{array}\right] =\left[\begin{array}{ll} 4 & -4\\ 0 & 0 \end{array}\right]$ $A(BC)=\left[\begin{array}{ll} 4 & 0\\ -3 & 5\\ 0 & 1 \end{array}\right]\left[\begin{array}{ll} 4 & -4\\ 0 & 0 \end{array}\right]$ $=\left[\begin{array}{ll} 16+0 & -16+0\\ -12+0 & 12+0\\ 0+0 & 0+0 \end{array}\right]$ $=\left[\begin{array}{ll} 16 & -16\\ -12 & 12\\ 0 & 0 \end{array}\right]$