Question

If $A = \begin{bmatrix} -k & 0 \\\ 0 & -k \end{bmatrix}, \, k \neq 0$, then the value of $m$ in $(A^T)^4 = mA$ is:

• A. $-k$
• B. $k^4$
• C. $-k^3$
• D. $\frac{1}{k}$

Answer 1

Answer: (C)

$-k^3$

Answer 2

The matrix $A$ is:

A = $\begin{bmatrix} -k & 0 \\\ 0 & -k \end{bmatrix}.$

Raise $A$ to the power of 4:

$A^4 = \begin{bmatrix} (-k)^4 & 0 \\\ 0 & (-k)^4 \end{bmatrix} = \begin{bmatrix} k^4 & 0 \\\ 0 & k^4 \end{bmatrix}.$

Now, solve for $m$ in:

$A^4 = mA \implies \begin{bmatrix} k^4 & 0 \\\ 0 & k^4 \end{bmatrix} = m \begin{bmatrix} -k & 0 \\\ 0 & -k \end{bmatrix}.$

Equating elements:
$k^4 = m(-k) \implies m = \frac{k^4}{-k} = -k^3.$

Thus, $m = -k^3$.