Question

If \( A = \begin{bmatrix} 1 & -2 & 3
-4 & 2 & 5 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 & 3
4 & 5
2 & 1 \end{bmatrix} \), then find \( AB \) and \( BA \).

Hint:

Matrix multiplication is not commutative, i.e., \( AB \neq BA \) in general.

Answer 1

Matrix multiplication is defined as: \[ AB = \begin{bmatrix} 1 & -2 & 3
-4 & 2 & 5 \end{bmatrix} \times \begin{bmatrix} 2 & 3
4 & 5
2 & 1 \end{bmatrix}. \] Calculating the elements: \[ AB = \begin{bmatrix} (1 \cdot 2 + (-2) \cdot 4 + 3 \cdot 2) & (1 \cdot 3 + (-2) \cdot 5 + 3 \cdot 1)
(-4 \cdot 2 + 2 \cdot 4 + 5 \cdot 2) & (-4 \cdot 3 + 2 \cdot 5 + 5 \cdot 1) \end{bmatrix}. \] \[ = \begin{bmatrix} (2 - 8 + 6) & (3 - 10 + 3)
(-8 + 8 + 10) & (-12 + 10 + 5) \end{bmatrix}. \] \[ = \begin{bmatrix} 0 & -4
10 & 3 \end{bmatrix}. \]