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  • Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 10

Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 10

Answers (1)

Answer: Hence proved A^{2}=0

Hint:

matrix multiplication is only possible, when the number of columns of first matrix is equal to the number of rows of second matrix.

Given: A=\left[\begin{array}{cc}a b & b^{2} \\ -a^{2} & -a b\end{array}\right]

Prove:A^{2}=0

Consider,

A^{2}=A A\\\\ A^{2}=\left[\begin{array}{cc}a b & b^{2} \\ -a^{2} & -a b\end{array}\right]\left[\begin{array}{cc}a b & b^{2} \\ -a^{2} & -a b\end{array}\right]\\\\\\ =\left[\begin{array}{cc}a b \times a b+b^{2} \times\left(-a^{2}\right) & a b \times b^{2}+b^{2} \times(-a b) \\ -a^{2} \times a b+(-a b) \times(-a)^{2} & -a^{2} \times b^{2}+(-a b) \times(-a b)\end{array}\right]\\\\ \\=\left[\begin{array}{cc}a^{2} b^{2}-a^{2} b^{2} & a b^{3}-a b^{3} \\ -a^{3} b+a^{3} b & -a^{2} b^{2}+a^{2} b^{2}\end{array}\right]=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]\\\\ A^{2}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]\\\\ A^{2}=0

Hence, proved.

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