Question

For a given matrix \(A = \begin{bmatrix} 4 & 4 \\ 0 & 4 \\ -4 & 4 \end{bmatrix}\), and \(A^T\) representing its transpose, which of the following options is the CORRECT representation of \(A^T A\)?

• A. \(\begin{bmatrix} 4 & 0 \\ 0 & -4 \end{bmatrix}\)
• B. \(\begin{bmatrix} 32 & 0 \\ 0 & 48 \end{bmatrix}\)
• C. \(\begin{bmatrix} 32 & 0 \\ 0 & 0 \\ 0 & 48 \end{bmatrix}\)
• D. \(\begin{bmatrix} 4 & 0 \\ 0 & 0 \\ 0 & -4 \end{bmatrix}\)

Hint:

Multiply the transpose of A by A; the result is a 2 by 2 symmetric matrix formed from dot products of the columns of A.

Answer 1

The Correct Option is B

Step 1: Write down the transpose of A:
The given matrix is \(A = \begin{bmatrix} 4 & 4 \\ 0 & 4 \\ -4 & 4 \end{bmatrix}\), which is a 3 by 2 matrix. Its transpose is obtained by writing the rows of A as the columns, giving \(A^T = \begin{bmatrix} 4 & 0 & -4 \\ 4 & 4 & 4 \end{bmatrix}\), which is a 2 by 3 matrix.
Step 2: Check the dimensions of the product:
Since \(A^T\) is 2 by 3 and A is 3 by 2, the product \(A^T A\) is a valid multiplication and results in a 2 by 2 matrix.
Step 3: Compute the first row of the product:
The entry in row 1 column 1 is the dot product of the first row of \(A^T\), which is \((4, 0, -4)\), with the first column of A, which is \((4, 0, -4)\). This gives \(4 \times 4 + 0 \times 0 + (-4) \times (-4) = 16 + 0 + 16 = 32\). The entry in row 1 column 2 is the dot product of \((4, 0, -4)\) with the second column of A, which is \((4, 4, 4)\). This gives \(4 \times 4 + 0 \times 4 + (-4) \times 4 = 16 + 0 - 16 = 0\).
Step 4: Compute the second row of the product:
The entry in row 2 column 1 is the dot product of the second row of \(A^T\), which is \((4, 4, 4)\), with the first column of A, which is \((4, 0, -4)\). This gives \(4 \times 4 + 4 \times 0 + 4 \times (-4) = 16 + 0 - 16 = 0\). The entry in row 2 column 2 is the dot product of \((4, 4, 4)\) with the second column of A, which is \((4, 4, 4)\). This gives \(4 \times 4 + 4 \times 4 + 4 \times 4 = 16 + 16 + 16 = 48\).
Step 5: Assemble the final matrix and check the options:
The product is \(A^T A = \begin{bmatrix} 32 & 0 \\ 0 & 48 \end{bmatrix}\), a 2 by 2 matrix. Option A gives the wrong dimension and wrong values. Option B matches exactly. Option C and option D are written as 3 by 2 matrices, which is an incorrect dimension for \(A^T A\), since a 2 by 3 matrix multiplied by a 3 by 2 matrix must give a 2 by 2 result.
Final Answer:
\[ \boxed{A^T A = \begin{bmatrix} 32 & 0 \\ 0 & 48 \end{bmatrix}} \]