Question

For what values of x and y are the following matrices equal?
A = $\left[\begin{array}{cc}2x+1& 2y\\ 0& y^2-5y\end{array}\right],B=\left[\begin{array}{cc}x+3& y^2+2\\ 0& -6\end{array}\right]$

Answer 1

We have, $A=\left[\begin{array}{cc}2x+1& 2y\\ 0& y^2-5y\end{array}\right],B=\left[\begin{array}{cc}x+3& y^2+2\\ 0& -6\end{array}\right]$

The corresponding elements of two equal matrices are equal.

$$\begin{gathered} \therefore 2x+1=x+3 \\ \Rightarrow 2x-x=3-1 \\ \Rightarrow x=2 \\ \mathrm{and}2y=y^2+2 \\ \Rightarrow y^2-2y+2=0 \\ \Rightarrow y=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \Rightarrow y=\frac{2\pm \sqrt{4-8}}{2} \\ \Rightarrow y=\frac{2\pm \sqrt{-4}}{2} \\ \mathrm{Also},y^2-5y=-6 \\ \Rightarrow y^2-5y+6=0 \\ \Rightarrow y^2-3y-2y+6=0 \\ \Rightarrow y\left(y-3\right)-2\left(y-3\right)=0 \\ \Rightarrow \left(y-3\right)\left(y-2\right)=0 \\ \Rightarrow y-3=0\mathrm{or}y-2=0 \\ \Rightarrow y=3\mathrm{or}y=2 \end{gathered}$$

Now, $3y=y^2+2\mathrm{and}{y}^2-5y=-6$ must have a common value.

Thus, A and B cannot be equal for any value of y.