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We are given the matrix \( A = \begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix} \) and asked to find \( (A^T)^2 \), where \( A^T \) is the transpose of matrix \( A \), and then squaring it.

### Step 1: Transpose of Matrix \( A \)
The transpose of matrix \( A \) is found by switching the rows and columns:
\[
A^T = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}
\]

### Step 2: Square of \( A^T \)
Now, we compute \( (A^T)^2 \), which means multiplying \( A^T \) by itself:
\[
(A^T)^2 = A^T \times A^T = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}
\]

Performing the matrix multiplication:
\[
\begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}
= \begin{bmatrix} (1 \times 1) + (2 \times -1) & (1 \times 2) + (2 \times 3) \\ (-1 \times 1) + (3 \times -1) & (-1 \times 2) + (3 \times 3) \end{bmatrix}
\]

Calculating each element:
\[
= \begin{bmatrix} 1 - 2 & 2 + 6 \\ -1 - 3 & -2 + 9 \end{bmatrix}
= \begin{bmatrix} -1 & 8 \\ -4 & 7 \end{bmatrix}
\]

### Step 3: Conclusion
We find that \( (A^T)^2 = \begin{bmatrix} -1 & 8 \\ -4 & 7 \end{bmatrix} \), which matches option **d**.

Would you like any further details or explanations?

Here are 5 related questions to expand your understanding:
1. How do you calculate the transpose of a matrix in general?
2. What are the properties of matrix multiplication?
3. How does the square of a matrix differ from multiplying two different matrices?
4. What is the importance of the transpose operation in linear algebra?
5. Can the process of squaring a matrix ever result in a matrix with zero entries?

**Tip:** When multiplying matrices, always ensure the dimensions are compatible, i.e., the number of columns in the first matrix must equal the number of rows in the second matrix.

If A = [ [1, -1], [2, 3] ], then (A^T)^2 = ?

Learn how to compute the square of a matrix transpose for A = [1, -1; 2, 3]. This problem explores key linear algebra concepts such as matrix transposition and multiplication, ideal for high school students in Grades 10-12.

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