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Determinants and Matrices — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsDeterminants and Matrices1 Mark
Question
If $A=\left[\begin{array}{cc}-3 & 2 \\ 2 & -4\end{array}\right], B=\left[\begin{array}{cc}1 & x \\ y & 0\end{array}\right]$ and $(A+B)(A-B)=A^2-B^2$, find $x$ and $y$.
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Answer

$\begin{aligned} & (A+B)(A-B)=A^2-B^2 \\ & A^2-A B+B A-B^2=A^2-B^2 \\ & -A B+B A=0 \\ & A B=B A\end{aligned}$

$\begin{aligned} & {\left[\begin{array}{cc}-3 & 2 \\ 2 & -4\end{array}\right]\left[\begin{array}{ll}1 & x \\ y & 0\end{array}\right]=\left[\begin{array}{ll}1 & x \\ y & 0\end{array}\right]\left[\begin{array}{cc}-3 & 2 \\ 2 & -4\end{array}\right]} \\ & {\left[\begin{array}{cc}-3+2 y & -3 x+0 \\ 2-4 y & 2 x+0\end{array}\right]=\left[\begin{array}{ll}-3+2 x & 2-4 x \\ -3 y+0 & 2 y+0\end{array}\right]}\end{aligned}$

By equality of matrices, we get 2 – 4x = -3x ∴ x = 2 and 2y = 2x y = x ∴ y = 2 ∴ x = 2, y = 2

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