Question

If a= [4,3]

[2,5] 2×2 matrix.

Find x and y such that A²-xA + y I = 0

Answer 1

$Given \begin{bmatrix} 4 & 3 \\ 2 & 5 \end{bmatrix}_{2\times 2}$

$A^2-xA+yI=0.....................................................(i)$

$A^2=A.A=\begin{bmatrix} 4 & 3\\ 2 & 5 \end{bmatrix}\begin{bmatrix} 4 & 3 \\ 2 & 5 \end{bmatrix}=\begin{bmatrix} 16+6 & 12+15 \\ 8+10 & 6+25 \end{bmatrix}$

$A^2=\begin{bmatrix} 22 & 27 \\ 18 & 31 \end{bmatrix}$

Value of A² and A put in equation (i). I is identity matrix.

$\begin{bmatrix} 22 & 27 \\ 18 & 31 \end{bmatrix}-x\begin{bmatrix} 4 & 3 \\ 2 & 5 \end{bmatrix}+y\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

$\begin{bmatrix} 22 & 27 \\ 18 & 31 \end{bmatrix}-\begin{bmatrix} 4x & 3x \\ 2x & 5x \end{bmatrix}+\begin{bmatrix} y & 0 \\ 0 & y \end{bmatrix}=\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

$\begin{bmatrix} 22-4x+y & 27-3x \\ 18-2x & 31-5x+y \end{bmatrix}=\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

$22-4xy+y=0.......................................................(ii)$

$27-3x=0$

$3x=27$

$x=9$

$18-2x=0$

$2x=18$

$x=9$

$31-5x+y=0$ putting the value of x in this equation,

$31-(5\times 9)+y=0$

$y=14$

$x=9,$ $y=14$