Find the products AB and BA to determine whether B is the multiplicative inverse of A.
-$$A = \left[ \begin{array} { r r r r }
1 & 0 & 0 & - 1 \
2 & 1 & 0 & 0 \
1 & 1 & 1 & - 2 \
0 & 0 & 0 & 1
\end{array} \right] , \quad B = \left[ \begin{array} { r r r r }
1 & 0 & 0 & 1 \
- 2 & 1 & 0 & - 2 \
1 & - 1 & 1 & 3 \
0 & 0 & 0 & 1
\end{array} \right]$$
A) $B = A ^ { - 1 }$
B) $B \neq A ^ { - 1 }$

Question

Find the products AB and BA to determine whether B is the multiplicative inverse of A.
-$$A = \left[ \begin{array} { r r r r } 
1 & 0 & 0 & - 1 \
2 & 1 & 0 & 0 \
1 & 1 & 1 & - 2 \
0 & 0 & 0 & 1
\end{array} \right] , \quad B = \left[ \begin{array} { r r r r } 
1 & 0 & 0 & 1 \
- 2 & 1 & 0 & - 2 \
1 & - 1 & 1 & 3 \
0 & 0 & 0 & 1
\end{array} \right]$$
A) $B = A ^ { - 1 }$
B) $B \neq A ^ { - 1 }$

Quizplus · Accepted Answer

\begin{align*}
AB &= \left[ \begin{array} { r r r r } 
1 & 0 & 0 & - 1 \
2 & 1 & 0 & 0 \
1 & 1 & 1 & - 2 \
0 & 0 & 0 & 1
\end{array} ight] \left[ \begin{array} { r r r r } 
1 & 0 & 0 & 1 \
- 2 & 1 & 0 & - 2 \
1 & - 1 & 1 & 3 \
0 & 0 & 0 & 1
\end{array} ight] \
&= \left[ \begin{array} { r r r r } 
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
\end{array} ight] = I
\end{align*}
\begin{align*}
BA &= \left[ \begin{array} { r r r r } 
1 & 0 & 0 & 1 \
- 2 & 1 & 0 & - 2 \
1 & - 1 & 1 & 3 \
0 & 0 & 0 & 1
\end{array} ight] \left[ \begin{array} { r r r r } 
1 & 0 & 0 & - 1 \
2 & 1 & 0 & 0 \
1 & 1 & 1 & - 2 \
0 & 0 & 0 & 1
\end{array} ight] \
&= \left[ \begin{array} { r r r r } 
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
\end{array} ight] = I
\end{align*}
Therefore, $A$ and $B$ are inverse matrices and $B = A^{-1}$, which means $B$ is the multiplicative inverse of $A$.

Find the Products AB and BA to Determine Whether B