Question

If \[ A=\begin{bmatrix} 2 & -3\\ -4 & 1 \end{bmatrix}, \] then \[ (A^T)^2+(12A)^T= \]

• A. \(5\begin{bmatrix}8 & 12\\ -9 & 5\end{bmatrix}\)
• B. \(5\begin{bmatrix}8 & -9\\ -12 & 5\end{bmatrix}\)
• C. \(\begin{bmatrix}40 & -45\\ 60 & 25\end{bmatrix}\)
• D. \(\begin{bmatrix}40 & -60\\ -45 & 25\end{bmatrix}\)

Hint:

Remember that: \[ (kA)^T=kA^T \] and matrix multiplication must always follow row-column multiplication rules.

Answer 1

The Correct Option is D

Step 1: Write down the transpose.
Given $A=\begin{bmatrix}2 & -3\\ -4 & 1\end{bmatrix}$, swapping rows and columns gives $A^T=\begin{bmatrix}2 & -4\\ -3 & 1\end{bmatrix}$.
Step 2: Compute $(A^T)^2$.
Multiply $A^T$ by itself: \[ (A^T)^2=\begin{bmatrix}2 & -4\\ -3 & 1\end{bmatrix}\begin{bmatrix}2 & -4\\ -3 & 1\end{bmatrix}=\begin{bmatrix}16 & -12\\ -9 & 13\end{bmatrix}. \]
Step 3: Use a transpose shortcut.
Since $(12A)^T=12A^T$, we get $12A^T=\begin{bmatrix}24 & -48\\ -36 & 12\end{bmatrix}$.
Step 4: Add the two matrices.
\[ (A^T)^2+(12A)^T=\begin{bmatrix}16+24 & -12-48\\ -9-36 & 13+12\end{bmatrix}. \]
Step 5: Simplify each entry.
This gives $\begin{bmatrix}40 & -60\\ -45 & 25\end{bmatrix}$.
Step 6: Match the option.
This is exactly option (4).
\[ \boxed{\begin{bmatrix}40 & -60\\ -45 & 25\end{bmatrix}} \]