Let A=left(begin{array}{rrr}1 & -1 & 0 0 & 1 & -1 0 & 0 & 1end{array}right)

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3 and 4 .Determinants and Matrices

hard

Let $A=\left(\begin{array}{rrr}1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1\end{array}\right)$ and $B=7 A^{20}-20 A^{7}+2 I$, where $I$ is an identity matrix of order $3 \times 3$ If $B=\left[b_{i j}\right]$, then $b_{13}$ is equal to $....$

$810$

$910$

$485$

$353$

(JEE MAIN-2021)

Solution

Let $A=\left(\begin{array}{ccc}0 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1\end{array}\right)=1+C$

Where $I=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right), C=\left(\begin{array}{ccc}0 & -1 & 0 \\ 0 & 0 & -1 \\ 0 & 0 & 0\end{array}\right)$

$C^{2}=\left(\begin{array}{lll}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)$

$C^{3}=\left(\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right), C^{4}=C^{5}=\ldots .$

$B=7 A^{20}-20 A^{7}+2 I$

$=7(1+c)^{20}-20(1+C)^{7}+2 I$

$\mathrm{B} 13=7 \times{ }^{20} \mathrm{C}_{2}-20 \times{ }^{7} \mathrm{C}_{2}=910$

Standard 12

Mathematics

If $A,B$are square matrices of order $3, A$ is non- singular and $AB = O$, then $B$ is a

normal

If $A = \left[ {\begin{array}{{20}{c}}\alpha &0\\1&1\end{array}} \right]$ and $B = \left[ {\begin{array}{{20}{c}}1&0\\5&1\end{array}} \right]$, then value of $\alpha $for which ${A^2} = B$, is

easy

(IIT-2003)

Find $x,$ if $[x-5-1]\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=O$

medium

If $x\left[\begin{array}{l}2 \\ 3\end{array}\right]+y\left[\begin{array}{l}-1 \\ 1\end{array}\right]=\left[\begin{array}{l}10 \\ 5\end{array}\right],$ find values of $\mathrm{x}$ and $\mathrm{y}$.

easy

If $R(t) = \left[ {\begin{array}{*{20}{c}}{\cos t}&{\sin t}\\{ – \sin t}&{\cos t}\end{array}} \right],$then $R(s).\,R(t) = $

easy

Let $A=\left(\begin{array}{rrr}1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1\end{array}\right)$ and $B=7 A^{20}-20 A^{7}+2 I$, where $I$ is an identity matrix of order $3 \times 3$ If $B=\left[b_{i j}\right]$, then $b_{13}$ is equal to $....$

Solution

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If $A = \left[ {\begin{array}{{20}{c}}\alpha &0\\1&1\end{array}} \right]$ and $B = \left[ {\begin{array}{{20}{c}}1&0\\5&1\end{array}} \right]$, then value of $\alpha $for which ${A^2} = B$, is