A = [x011],B=[40y1] and C=[40x1]\begin{bmatrix*}[r] x & 0 \\ 1 & 1 \end{bmatrix*}, B = \begin{bmatrix*}[r] 4 & 0 \\ y & 1 \end{bmatrix*}\text{ and } C = \begin{bmatrix*}[r] 4 & 0 \\ x & 1 \end{bmatrix*}[x101],B=[4y01] and C=[4x01].
Find the values of x and y, if AB = C.
⇒AB=C⇒[x011][40y1]=[40x1]⇒[x×4+0×yx×0+0×11×4+1×y1×0+1×1]=[40x1]⇒[4x+00+04+y0+1]=[40x1]⇒[4x04+y1]=[40x1]⇒4x=4 and 4+y=x⇒x=44 and 4+y=x⇒x=1 and 4+y=1⇒x=1 and y=1−4=−3.\phantom{\Rightarrow} AB = C \\[1em] \Rightarrow \begin{bmatrix*}[r] x & 0 \\ 1 & 1 \end{bmatrix*}\begin{bmatrix*}[r] 4 & 0 \\ y & 1 \end{bmatrix*} = \begin{bmatrix*}[r] 4 & 0 \\ x & 1 \end{bmatrix*} \\[1em] \Rightarrow \begin{bmatrix*}[r] x \times 4 + 0 \times y & x \times 0 + 0 \times 1 \\ 1 \times 4 + 1 \times y & 1 \times 0 + 1 \times 1 \end{bmatrix*} = \begin{bmatrix*}[r] 4 & 0 \\ x & 1 \end{bmatrix*} \\[1em] \Rightarrow \begin{bmatrix*}[r] 4x + 0 & 0 + 0 \\ 4 + y & 0 + 1 \end{bmatrix*} = \begin{bmatrix*}[r] 4 & 0 \\ x & 1 \end{bmatrix*} \\[1em] \Rightarrow \begin{bmatrix*}[r] 4x & 0 \\ 4 + y & 1 \end{bmatrix*} = \begin{bmatrix*}[r] 4 & 0 \\ x & 1 \end{bmatrix*} \\[1em] \Rightarrow 4x = 4 \text{ and } 4 + y = x \\[1em] \Rightarrow x = \dfrac{4}{4} \text{ and } 4 + y = x \\[1em] \Rightarrow x = 1 \text{ and } 4 + y = 1 \\[1em] \Rightarrow x = 1 \text{ and } y = 1 - 4 = -3.⇒AB=C⇒[x101][4y01]=[4x01]⇒[x×4+0×y1×4+1×yx×0+0×11×0+1×1]=[4x01]⇒[4x+04+y0+00+1]=[4x01]⇒[4x4+y01]=[4x01]⇒4x=4 and 4+y=x⇒x=44 and 4+y=x⇒x=1 and 4+y=1⇒x=1 and y=1−4=−3.
Hence, x = 1 and y = -3.
