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  • Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 53 math

Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 53 math

Answers (1)

Answer:

A^{2}-5 A-14 I=\left[\begin{array}{cc}0 & 0 \\ 0 & 0\end{array}\right]=0\ \ and \ \ A^{3}=\left[\begin{array}{cc}187 & -195 \\ -156 & 148\end{array}\right]

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}3 & -5 \\ -4 & 2\end{array}\right]

First, we solve this   A^{2}-5 A-14 I

Where I is an identity matrix

Then,

\\\\ \left[\begin{array}{cc}3 & -5 \\ -4 & 2\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ -4 & 2\end{array}\right]-5\left[\begin{array}{cc}3 & -5 \\ -4 & 2\end{array}\right]-14\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]\\\\ \\=\left[\begin{array}{cc}9+20 & -15-10 \\ -12-8 & 20+4\end{array}\right]-\left[\begin{array}{cc}15 & -25 \\ -20 & 10\end{array}\right]-\left[\begin{array}{cc}14 & 0 \\ 0 & 14\end{array}\right]

\\\\ =\left[\begin{array}{cc} 29 & -25 \\ -20 & 24 \end{array}\right]-\left[\begin{array}{cc} 15 & -25 \\ -20 & 10 \end{array}\right]-\left[\begin{array}{cc} 14 & 0 \\ 0 & 14 \end{array}\right]\\\\\\ =\left[\begin{array}{cc} 29-15-14 & -25+25-0 \\ -20+20-0 & 24-10-14 \end{array}\right]\\\\\\ =\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]=0\\\\\\ A^{2}-5 A-14 I=0\\\\

Now,

\\\\ A^{2}-5 A-14 I=0\\\\ A^{2}=5 A+14 I\\\\ A^{3}=A^{2} A=(5 A+14 I) A\\\\ A^{3}=5 A^{2}+14 A\\\\ A^{3}=A^{2} A=5 A^{2}+14 A

As we know AI=A

\\\\ A^{3}=5\left[\begin{array}{cc} 29 & -25 \\ -20 & 24 \end{array}\right]+14\left[\begin{array}{cc} 3 & -5 \\ -4 & 2 \end{array}\right]\\\\\\ A^{3}=\left[\begin{array}{cc} 145 & -125 \\ -100 & 120 \end{array}\right]+\left[\begin{array}{cc} 42 & -70 \\ -56 & 28 \end{array}\right]\\\\\\ A^{3}=\left[\begin{array}{cc} 145+42 & -125-70 \\ -100-56 & 120+28 \end{array}\right]\\\\\\ A^{3}=\left[\begin{array}{cc} 187 & -195 \\ -156 & 148 \end{array}\right]

Posted by

infoexpert22

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