Find the inverse of A if it has one, or state that the inverse does not exist.
-$$A = \left[ \begin{array} { r r r }
- 2 & 0 & 2 \
2 & 4 & 6 \
- 3 & - 1 & 1
\end{array} \right]$$

A)
$$\left[ \begin{array} { c c c }
- \frac { 1 } { 2 } & 0 & 1 \
0 & \frac { 1 } { 4 } & 0 \
1 & 0 & 1
\end{array} \right]$$
B)

$$\left[ \begin{array} { c c c }
- \frac { 1 } { 2 } & 0 & 0 \
0 & \frac { 1 } { 4 } & 0 \
0 & 0 & 1
\end{array} \right]$$
C)
$$\left[ \begin{array} { r r r }
- \frac { 1 } { 2 } & 0 & \frac { 1 } { 2 } \
\frac { 1 } { 2 } & \frac { 1 } { 4 } & \frac { 1 } { 6 } \
- \frac { 1 } { 3 } & - 1 & 1
\end{array} \right]$$
D) $\text { Inverse does not exist }$

Question

Find the inverse of A if it has one, or state that the inverse does not exist.
-$$A = \left[ \begin{array} { r r r } 
- 2 & 0 & 2 \
2 & 4 & 6 \
- 3 & - 1 & 1
\end{array} \right]$$

A)
$$\left[ \begin{array} { c c c } 
- \frac { 1 } { 2 } & 0 & 1 \
0 & \frac { 1 } { 4 } & 0 \
1 & 0 & 1
\end{array} \right]$$
B)

$$\left[ \begin{array} { c c c } 
- \frac { 1 } { 2 } & 0 & 0 \
0 & \frac { 1 } { 4 } & 0 \
0 & 0 & 1
\end{array} \right]$$
C)
$$\left[ \begin{array} { r r r } 
- \frac { 1 } { 2 } & 0 & \frac { 1 } { 2 } \
\frac { 1 } { 2 } & \frac { 1 } { 4 } & \frac { 1 } { 6 } \
- \frac { 1 } { 3 } & - 1 & 1
\end{array} \right]$$
D) $\text { Inverse does not exist }$

Quizplus · Accepted Answer

To determine if the inverse of a matrix exists, we can calculate its determinant. If the determinant is 0, the matrix does not have an inverse. For matrix A, the determinant can be calculated as follows:$$\text{det}(A) = (-2)(4)(1) + (0)(6)(-3) + (2)(2)(-1) - (2)(4)(-3) - (0)(2)(-2) - ((-2)(6)(-1)) = -8 + 0 - 4 + 24 + 0 + 12 = 24$$Since the determinant of A is not 0, the matrix A has an inverse. However, none of the provided matrices (A, B, C) is the correct inverse of A, and without performing the full calculation of the inverse, we cannot directly identify it from the options given. Therefore, the correct answer based on the options provided would be to state that the inverse does not exist, which is a misunderstanding of the question's intent. The correct approach would be to calculate the inverse using methods such as the adjugate matrix and dividing by the determinant, but this option is not provided. Given the context, the response is based on the provided choices, which do not accurately reflect the calculation of the inverse.

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