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  • Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 46

Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 46

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Answer:  Hence, prove A^{2}-7 A+10 I_{3}=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}{lll} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{array}\right]

Prove: A^{2}-7 A+10 I_{3}=0

I_3 is identity matrix of size 3
 

I_{3}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\\\\\\ \mathrm{LHS}=A^{2}-7 A+10 I_{3}\\\\ =\left[\begin{array}{lll} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{array}\right]\left[\begin{array}{lll} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{array}\right]-7\left[\begin{array}{lll} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{array}\right]+10\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\\

\begin {array} {ll} =\left[\begin{array}{ccc} 9+2+0 & 6+8+0 & 0+0+0 \\ 3+4+0 & 2+16+0 & 0+0+0 \\ 0+0+0 & 0+0+0 & 0+0+25 \end{array}\right]-\left[\begin{array}{ccc} 21 & 14 & 0 \\ 7 & 28 & 0 \\ 0 & 0 & 35 \end{array}\right]+\left[\begin{array}{ccc} 10 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 10 \end{array}\right]\\\\\\ =\left[\begin{array}{ccc} 11 & 14 & 0 \\ 7 & 18 & 0 \\ 0 & 0 & 25 \end{array}\right]-\left[\begin{array}{ccc} 21 & 14 & 0 \\ 7 & 28 & 0 \\ 0 & 0 & 35 \end{array}\right]+\left[\begin{array}{ccc} 10 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 10 \end{array}\right]\\\\\\ =\left[\begin{array}{ccc} 11-21+10 & 14-14+0 & 0-0+0 \\ 7-7+0 & 18-28+10 & 0-0+0 \\ 0-0+0 & 0-0+0 & 25-35+10 \end{array}\right]\\\\\\ =\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]=0\\ \end{}

Hence, A^{2}-7 A+10 I_{3}=0

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