Question:
If
\[
A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix}, \quad
B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix}
\]
Then find \( AB \) and \( BA \).
If
\[ A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix} \]Then find \( AB \) and \( BA \).
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Matrix multiplication is not commutative, i.e., \( AB \neq BA \) in general.
Updated On: Mar 3, 2025
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Solution and Explanation
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:
\[ AB = \begin{bmatrix} (1 \cdot 2 + (-2) \cdot 4 + 3 \cdot 2) & (1 \cdot 3 + (-2) \cdot 5 + 3 \cdot 1) \\ (-4 \cdot 2 + 2 \cdot 4 + 5 \cdot 2) & (-4 \cdot 3 + 2 \cdot 5 + 5 \cdot 1) \end{bmatrix}. \] \[ = \begin{bmatrix} (2 - 8 + 6) & (3 - 10 + 3) \\ (-8 + 8 + 10) & (-12 + 10 + 5) \end{bmatrix}. \] \[ = \begin{bmatrix} 0 & -4 \\ 10 & 3 \end{bmatrix}. \]Was this answer helpful?
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