Question

If \[ A = \begin{bmatrix} 1 & -1 & 2 \\ -2 & 3 & -3 \\ 4 & -4 & 5 \end{bmatrix} \] and \( A^T \) represents the transpose of \( A \), then calculate \( AA^T - A - A^T \).

• A. \(\begin{bmatrix}4 & 8 & 12 \\ 8 & 16 & -28 \\ 12 & -28 & 47\end{bmatrix}\)
• B. \(\begin{bmatrix}4 & -8 & 12 \\ -8 & 16 & -28 \\ 12 & -28 & 47\end{bmatrix}\)
• C. \(\begin{bmatrix}4 & -8 & 12 \\ -8 & 16 & 28 \\ 12 & 28 & 47\end{bmatrix}\)
• D. \(\begin{bmatrix}4 & 8 & -12 \\ 8 & 16 & 28 \\ -12 & 28 & 47\end{bmatrix}\)

Hint:

When calculating expressions involving matrices and their transpose, carefully compute each term element-wise.

Answer 1

The Correct Option is B

Step 1: Calculate \( A^T \)
\[ A^T = \begin{bmatrix} 1 & -2 & 4 \\ -1 & 3 & -4 \\ 2 & -3 & 5 \end{bmatrix} \] Step 2: Calculate \( AA^T \)
\[ AA^T = \begin{bmatrix} 1 & -1 & 2 \\ -2 & 3 & -3 \\ 4 & -4 & 5 \end{bmatrix} \begin{bmatrix} 1 & -2 & 4 \\ -1 & 3 & -4 \\ 2 & -3 & 5 \end{bmatrix} \] Calculate element-wise: - \( (1,1) = 1*1 + (-1)*(-1) + 2*2 = 1 + 1 + 4 = 6 \)
- \( (1,2) = 1*(-2) + (-1)*3 + 2*(-3) = -2 - 3 - 6 = -11 \)
- \( (1,3) = 1*4 + (-1)*(-4) + 2*5 = 4 + 4 + 10 = 18 \)
- \( (2,1) = -2*1 + 3*(-1) + (-3)*2 = -2 - 3 - 6 = -11 \)
- \( (2,2) = -2*(-2) + 3*3 + (-3)*(-3) = 4 + 9 + 9 = 22 \)
- \( (2,3) = -2*4 + 3*(-4) + (-3)*5 = -8 - 12 - 15 = -35 \)
- \( (3,1) = 4*1 + (-4)*(-1) + 5*2 = 4 + 4 + 10 = 18 \)
- \( (3,2) = 4*(-2) + (-4)*3 + 5*(-3) = -8 - 12 - 15 = -35 \)
- \( (3,3) = 4*4 + (-4)*(-4) + 5*5 = 16 + 16 + 25 = 57 \)
So, \[ AA^T = \begin{bmatrix} 6 & -11 & 18 \\ -11 & 22 & -35 \\ 18 & -35 & 57 \end{bmatrix} \] Step 3: Calculate \( AA^T - A - A^T \)
\[ AA^T - A - A^T = \begin{bmatrix} 6 & -11 & 18 \\ -11 & 22 & -35 \\ 18 & -35 & 57 \end{bmatrix} - \begin{bmatrix} 1 & -1 & 2 \\ -2 & 3 & -3 \\ 4 & -4 & 5 \end{bmatrix} - \begin{bmatrix} 1 & -2 & 4
-1 & 3 & -4
2 & -3 & 5 \end{bmatrix} \] Calculate each element: - \( (1,1): 6 - 1 - 1 = 4 \)
- \( (1,2): -11 - (-1) - (-2) = -11 + 1 + 2 = -8 \)
- \( (1,3): 18 - 2 - 4 = 12 \)
- \( (2,1): -11 - (-2) - (-1) = -11 + 2 + 1 = -8 \)
- \( (2,2): 22 - 3 - 3 = 16 \)
- \( (2,3): -35 - (-3) - (-4) = -35 + 3 + 4 = -28 \)
- \( (3,1): 18 - 4 - 2 = 12 \)
- \( (3,2): -35 - (-4) - (-3) = -35 + 4 + 3 = -28 \)
- \( (3,3): 57 - 5 - 5 = 47 \)
Thus, \[ AA^T - A - A^T = \begin{bmatrix} 4 & -8 & 12 \\ -8 & 16 & -28 \\ 12 & -28 & 47 \end{bmatrix} \]