Question

please solve this ,,,,,,,,,,,,, ​

Answer 1

$\Large{\underline{\underline{\mathfrak{\bf{Solution}}}}}$

$\Large{\underline{\mathfrak{\bf{\pink{Given}}}}}$

$\mapsto\sf{\:A\:=\:\left[\begin{array}{c c} 2 3 \\ 1 2 \end{array}\right]}$

• A² = xA + yI .........(1)

$\Large{\underline{\mathfrak{\bf{\pink{Find}}}}}$

• Value of x and y

$\Large{\underline{\underline{\mathfrak{\bf{Explanation}}}}}$

Here, I is a Identity matrix

So, it will be

$\mapsto\sf{\:I\:=\:\left[\begin{array}{c c} 1 0 \\ 0 1 \end{array}\right]}$

First Calculate:-

• value of A²

$\mapsto\sf{\:A^2\:=\:A\times A}$

Keep matrix A ,

$\mapsto\sf{\:A^2\:=\:\left[\begin{array}{c c} 2 3 \\ 1 2 \end{array}\right]\times \left[\begin{array}{c c} 2 3 \\ 1 2 \end{array}\right]} \\ \\ \\ \mapsto\sf{\:A^2\:=\:\left[\begin{array}{c c}2(2)+3(1) 2(3)+3(2) \\ 1(2)+2(1) 1(3)+2(2) \end{array}\right]} \\ \\ \\ \mapsto\sf{\:A^2\:=\:\left[\begin{array}{c c} 4+3 6+6 \\ 2+2 3+4 \end{array}\right]} \\ \\ \\ \mapsto\sf{\:A^2\:=\:\left[\begin{array}{c c} 7 12 \\ 4 7\end{array}\right]}$

Keep value in equ(1) of A² , I and A

$\mapsto\sf{\:\left[\begin{array}{c c} 7 12 \\ 4 7\end{array}\right]\:=\:x\left[\begin{array}{c c} 2 3 \\ 1 2\end{array}\right]\:+\:y\left[\begin{array}{c c} 1 0 \\ 0 1\end{array}\right]}$

$\mapsto\sf{\:\left[\begin{array}{c c} 7 12 \\ 4 7\end{array}\right]\:=\:\left[\begin{array}{c c} 2x+1y 3x+0 \\ 1x+0 2x+1y\end{array}\right]}$

compare both side,

• 2x + y = 7 .........(2)
• 3x = 12..............(3)
• => x = 12/3
• => x = 4
• x = 4 .............(4)
• 2x + y = 7 .........(5)

Keep value of x in equ(5),

$\mapsto\sf{\:2\times 4\:+\:y\:=\:7} \\ \mapsto\sf{\:y\:=\:7-8} \\ \mapsto\sf{\:y\:=\:-1}$

Thus:-

• Value of x = 4
• Value of y = -1