Question

If \[ A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -1 \end{pmatrix} \] then \( A^4 A^{-1} = \)

• A. \( \begin{pmatrix} 8 & 0 & 0\\ 0 & -8 & 0 \\ 0 & 0 & -1 \end{pmatrix} \)
• B. \( \begin{pmatrix} 8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)
• C. \( \begin{pmatrix} \frac{1}{2} & 0 & 0 \\ 0 & -\frac{1}{2} & 0 \\ 0 & 0 & -1 \end{pmatrix} \)
• D. \( \begin{pmatrix} -4 & 0 & 0\\0 & 4 & 0 \\ 0 & 0 & -1 \end{pmatrix} \)

Hint:

When computing powers and inverses of diagonal matrices, raise or invert each diagonal element independently.

Answer 1

The Correct Option is A

Step 1: Compute \( A^4 \).
Since \( A = \begin{pmatrix} 2 & 0 & 0 \\0 & -2 & 0 \\ 0 & 0 & -1 \end{pmatrix} \), we compute \( A^4 \) by raising each diagonal element to the power of 4: \[ A^4 = \begin{pmatrix} 2^4 & 0 & 0 \\ 0 & (-2)^4 & 0 \\ 0 & 0 & (-1)^4 \end{pmatrix} = \begin{pmatrix} 16 & 0 & 0 \\ 0 & 16 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
Step 2: Compute \( A^{-1} \).
The inverse of \( A \) is obtained by taking the reciprocal of each diagonal element: \[ A^{-1} = \begin{pmatrix} \frac{1}{2} & 0 & 0 \\ 0 & -\frac{1}{2} & 0 \\ 0 & 0 & -1 \end{pmatrix} \]
Step 3: Multiply \( A^4 \) and \( A^{-1} \).
Now, multiply \( A^4 \) and \( A^{-1} \): \[ A^4 A^{-1} = \begin{pmatrix} 16 & 0 & 0 \\ 0 & 16 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \frac{1}{2} & 0 & 0 \\ 0 & -\frac{1}{2} & 0 \\ 0 & 0 & -1 \end{pmatrix} = \begin{pmatrix} 8 & 0 & 0 \\ 0 & -8 & 0 \\ 0 & 0 & -1 \end{pmatrix} \]
Step 4: Conclusion.
Thus, the result is \( A^4 A^{-1} = \begin{pmatrix} 8 & 0 & 0 \\ 0 & -8 & 0 \\ 0 & 0 & -1 \end{pmatrix} \), corresponding to option (A).