Given, A = $$\begin{bmatrix} 3 & 1 \\ 5 & 3 \end{bmatrix}$$ and B = $$\begin{bmatrix} -1 & a \\ 3 & -5 \end{bmatrix}$$ and product AB = $$\begin{bmatrix} b & 7 \\ 4 & 5 \end{bmatrix}$$. Find the values of ‘a’ and ‘b’.
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Given,
A = $$\begin{bmatrix} 3 & 1 \\ 5 & 3 \end{bmatrix}$$
B = $$\begin{bmatrix} -1 & a \\ 3 & -5 \end{bmatrix}$$
AB = $$\begin{bmatrix} b & 7 \\ 4 & 5 \end{bmatrix}$$
Solving,
$$\Rightarrow AB = \begin{bmatrix} 3 & 1 \\ 5 & 3 \end{bmatrix} \times \begin{bmatrix} -1 & a \\ 3 & -5 \end{bmatrix} \\[1em] \Rightarrow AB = \begin{bmatrix} 3(-1) + 1(3) & 3(a) + 1(-5) \\ 5(-1) + 3(3) & 5(a) + 3(-5) \end{bmatrix} \\[1em] \Rightarrow AB = \begin{bmatrix} -3 + 3 & 3a - 5 \\ -5 + 9 & 5a - 15 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix} b & 7 \\ 4 & 5 \end{bmatrix} = \begin{bmatrix} 0 & 3a - 5 \\ 4 & 5a - 15 \end{bmatrix} \\[1em]$$
Comparing corresponding elements,
∴ b = 0
⇒ 3a - 5 = 7
⇒ 3a = 7 + 5
⇒ 3a = 12
⇒ a = $\frac{12}{3}$
⇒ a = 4
Hence, a = 4 and b = 0.