Question

If \(A = \begin{bmatrix} 4&5&2\\ 3&-1&7\end{bmatrix}\), then the sum of the elements of the matrix AA^T is:

• A. 156
• B. 164
• C. 146
• D. 136

Answer 1

The Correct Option is C

To find the sum of the elements of the matrix \(AA^T\), we first need to calculate \(A^T\), the transpose of matrix \(A\). Given the matrix \(A = \begin{bmatrix} 4 & 5 & 2 \\ 3 & -1 & 7 \end{bmatrix}\), its transpose \(A^T\) is \(\begin{bmatrix} 4 & 3 \\ 5 & -1 \\ 2 & 7 \end{bmatrix}\).

Next, compute the product \(AA^T\):

\[AA^T = \begin{bmatrix} 4 & 5 & 2 \\ 3 & -1 & 7 \end{bmatrix} \begin{bmatrix} 4 & 3 \\ 5 & -1 \\ 2 & 7 \end{bmatrix}\]

Perform the multiplication:

\[AA^T = \begin{bmatrix} (4 \times 4 + 5 \times 5 + 2 \times 2) & (4 \times 3 + 5 \times -1 + 2 \times 7) \\ (3 \times 4 + -1 \times 5 + 7 \times 2) & (3 \times 3 + -1 \times -1 + 7 \times 7) \end{bmatrix}\]

Simplify each element:

\[AA^T = \begin{bmatrix} 16 + 25 + 4 & 12 - 5 + 14 \\ 12 - 5 + 14 & 9 + 1 + 49 \end{bmatrix}\]

\[AA^T = \begin{bmatrix} 45 & 21 \\ 21 & 59 \end{bmatrix}\]

The sum of the elements in \(AA^T\) is calculated as follows:

\[45 + 21 + 21 + 59 = 146\]

Therefore, the sum of the elements of the matrix \(AA^T\) is 146.