Find the inverse of A if it has one, or state that the inverse does not exist.
-$$A = \left[ \begin{array} { r r r }
6 & 0 & 0 \
0 & - 6 & 0 \
6 & 0 & 6
\end{array} \right]$$
A)
$$\left[ \begin{array} { c c c }
\frac { 1 } { 6 } & 0 & 0 \
0 & - \frac { 1 } { 6 } & 0 \
- \frac { 1 } { 6 } & 0 & \frac { 1 } { 6 }
\end{array} \right]$$
B) Inverse does not exist
C)
$$\left[ \begin{array} { r r r }
- 6 & 0 & 0 \
0 & 6 & 0 \
- 6 & 0 & - 6
\end{array} \right]$$
D)
$$\left[ \begin{array} { c c c }
- \frac { 1 } { 6 } & 0 & 0 \
0 & \frac { 1 } { 6 } & 0 \
\frac { 1 } { 6 } & 0 & - \frac { 1 } { 6 }
\end{array} \right]$$

Question

Find the inverse of A if it has one, or state that the inverse does not exist.
-$$A = \left[ \begin{array} { r r r } 
6 & 0 & 0 \
0 & - 6 & 0 \
6 & 0 & 6
\end{array} \right]$$
A)
$$\left[ \begin{array} { c c c } 
\frac { 1 } { 6 } & 0 & 0 \
0 & - \frac { 1 } { 6 } & 0 \
- \frac { 1 } { 6 } & 0 & \frac { 1 } { 6 }
\end{array} \right]$$
B) Inverse does not exist
C)
$$\left[ \begin{array} { r r r } 
- 6 & 0 & 0 \
0 & 6 & 0 \
- 6 & 0 & - 6
\end{array} \right]$$
D)
$$\left[ \begin{array} { c c c } 
- \frac { 1 } { 6 } & 0 & 0 \
0 & \frac { 1 } { 6 } & 0 \
\frac { 1 } { 6 } & 0 & - \frac { 1 } { 6 }
\end{array} \right]$$

Quizplus · Accepted Answer

The inverse of a diagonal matrix can be found by taking the reciprocal of each non-zero diagonal element. The third row of matrix A is not purely diagonal, but the matrix can be considered block diagonal with the lower right element being part of a 1x1 block. The inverse of each diagonal element is $\frac{1}{6}$, $-\frac{1}{6}$, and $\frac{1}{6}$ respectively. The off-diagonal element in the inverse matrix accounts for the interaction between the first and third rows of the original matrix, leading to the correct inverse as given in option A.

Find the Inverse of a If It Has One, or State