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

The Correct Option is C

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, \(m = -k^3\).