Determine whether matrix B is the inverse of matrix A by finding the product AB.
-$A = \left[ \begin{array} { l l } - 5 & 1 \ - 7 & 1 \end{array} \right] , B = \left[ \begin{array} { l l } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \ \frac { 7 } { 2 } & - \frac { 5 } { 2 } \end{array} \right]$
A) $B \neq A ^ { - 1 }$
B) $B = A ^ { - 1 }$

Question

Quizplus · Accepted Answer

To determine if $B$ is the inverse of $A$, we calculate the product $AB$. The product $AB$ is calculated as follows:$$AB = \left[ \begin{array} { l l } - 5 & 1 \ - 7 & 1 \end{array} \right] \left[ \begin{array} { l l } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \ \frac { 7 } { 2 } & - \frac { 5 } { 2 } \end{array} \right]$$$$= \left[ \begin{array} { l l } - 5 \cdot \frac { 1 } { 2 } + 1 \cdot \frac { 7 } { 2 } & - 5 \cdot - \frac { 1 } { 2 } + 1 \cdot - \frac { 5 } { 2 } \ - 7 \cdot \frac { 1 } { 2 } + 1 \cdot \frac { 7 } { 2 } & - 7 \cdot - \frac { 1 } { 2 } + 1 \cdot - \frac { 5 } { 2 } \end{array} \right]$$$$= \left[ \begin{array} { l l } - \frac { 5 } { 2 } + \frac { 7 } { 2 } & \frac { 5 } { 2 } - \frac { 5 } { 2 } \ - \frac { 7 } { 2 } + \frac { 7 } { 2 } & \frac { 7 } { 2 } - \frac { 5 } { 2 } \end{array} \right]$$$$= \left[ \begin{array} { l l } 1 & 0 \ 0 & 1 \end{array} \right]$$Since the product $AB$ equals the identity matrix, $B$ is indeed the inverse of $A$.

Determine Whether Matrix B Is the Inverse of Matrix a by Finding