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  • Please solve RD Sharma class 12 chapter Algebra of matrices exercise 4.3 question 31 maths textbook solution

Please solve RD Sharma class 12 chapter Algebra of matrices exercise 4.3 question 31 maths textbook solution

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Answer: Hence, proved A^{3}-4 A^{2}+A=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}{ll}2 & 3 \\ 1 & 2\end{array}\right]

Prove:  A^{3}-4 A^{2}+A=0

Consider,A^{2}=A A

A^{2}=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]=\left[\begin{array}{ll}4+3 & 6+6 \\ 2+2 & 3+4\end{array}\right]=\left[\begin{array}{cc}7 & 12 \\ 4 & 7\end{array}\right] \\\\ A^{3}=A^{2} A\\\\ A^{2} A=\left[\begin{array}{cc}7 & 12 \\ 4 & 7\end{array}\right]\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]\\\\ A^{3}=\left[\begin{array}{cc}14+12 & 21+24 \\ 8+7 & 12+14\end{array}\right]=\left[\begin{array}{ll}26 & 45 \\ 15 & 26\end{array}\right]

Now putting value of A^{3}, A^{2} and A in the equation A^{3}-4 A^{2}+Awe get,

=\left[\begin{array}{cc}26 & 45 \\ 15 & 26\end{array}\right]-4\left[\begin{array}{cc}7 & 12 \\ 4 & 7\end{array}\right]+\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]

=\left[\begin{array}{ll} 26 & 45 \\ 15 & 26 \end{array}\right]-\left[\begin{array}{ll} 28 & 48 \\ 16 & 28 \end{array}\right]+\left[\begin{array}{ll} 2 & 3 \\ 1 & 2 \end{array}\right] \\ =\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \\\\ =0 \\ \text { So, } A^{3}-4 A^{2}+A=0

 

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