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

To solve for \( m \) in the equation \((A^T)^4 = mA\), we start by evaluating the properties of the given matrix \( A \).

Matrix \( A \) is given by: \[ A = \begin{bmatrix} -k & 0 \\ 0 & -k \end{bmatrix} \]

Step 1: Transpose of Matrix \( A \)

The transpose of a diagonal matrix \( A \) is simply the matrix itself, since transposing changes component (i, j) to (j, i), both being equal for diagonal elements. Thus:

\( A^T = \begin{bmatrix} -k & 0 \\ 0 & -k \end{bmatrix} = A \)

Step 2: Calculating \((A^T)^4\)

Since \( A^T = A \), we have:

\((A^T)^4 = A^4\)

To compute \( A^4 \), we first need \( A^2 \):

\( A^2 = A \cdot A = \begin{bmatrix} -k & 0 \\ 0 & -k \end{bmatrix} \cdot \begin{bmatrix} -k & 0 \\ 0 & -k \end{bmatrix} = \begin{bmatrix} k^2 & 0 \\ 0 & k^2 \end{bmatrix} \)

Then, \( A^4 = A^2 \cdot A^2 \):

\( A^4 = \begin{bmatrix} k^2 & 0 \\ 0 & k^2 \end{bmatrix} \cdot \begin{bmatrix} k^2 & 0 \\ 0 & k^2 \end{bmatrix} = \begin{bmatrix} k^4 & 0 \\ 0 & k^4 \end{bmatrix} \)

Step 3: Solve for \( m \) in \((A^T)^4 = mA\)

We equate \((A^T)^4\) and \( mA \):

\( \begin{bmatrix} k^4 & 0 \\ 0 & k^4 \end{bmatrix} = m \begin{bmatrix} -k & 0 \\ 0 & -k \end{bmatrix} \)

This gives two scalar equations:

\( k^4 = m(-k) \)

Simplifying, we get:

\( k^4 = -mk \)

Solving for \( m \), we divide both sides by \(-k\):

\( m = -k^3 \)

Thus, the value of \( m \) is \(-k^3\).

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, \(m = -k^3\).