QQuestion
If A = \begin{bmatrix}3 & 1 \\ 5 & 3\end{bmatrix}, B = \begin{bmatrix}-1 & \mathrm{a} \\ 3 & -5\end{bmatrix} and AB = \begin{bmatrix}\mathrm{b} & 7 \\ 4 & 5\end{bmatrix}, find the values of a and b.
ICSEClass XMathematicsMatrices04 Mark2026 Hard
✓Answer
Given:
A = \begin{bmatrix}3 & 1 \\ 5 & 3\end{bmatrix} and B = \begin{bmatrix}-1 & \mathrm{a} \\ 3 & -5\end{bmatrix}
AB = \begin{bmatrix}3(-1)+1(3) & 3\mathrm{a}+1(-5) \\ 5(-1)+3(3) & 5\mathrm{a}+3(-5)\end{bmatrix}
⇒ AB = \begin{bmatrix}0 & 3\mathrm{a}-5 \\ 4 & 5\mathrm{a}-15\end{bmatrix}
But AB = \begin{bmatrix}\mathrm{b} & 7 \\ 4 & 5\end{bmatrix}
⇒ \begin{bmatrix}0 & 3\mathrm{a}-5 \\ 4 & 5\mathrm{a}-15\end{bmatrix} = \begin{bmatrix}\mathrm{b} & 7 \\ 4 & 5\end{bmatrix}
Comparing corresponding elements,
b = 0
and 3a − 5 = 7
⇒ 3a = 12
⇒ a = 4
Verification using the bottom-right element:
5a − 15 = 5
⇒ 5(4) − 15 = 5
⇒ 20 − 15 = 5
Therefore, a = 4 and b = 0.
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