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For the matrix A=[[3 ,1], [7, 5]] , find...

For the matrix `A=[[3 ,1], [7, 5]]` , find `x` and `y` so that `A^2+x I=y Adot`

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`A=[[3,1],[7,5]]`
`A^2=AA=[[3,1],[7,5]][[3,1],[7,5]]` `=[[16,8],[56,32]]`
Now, `A^2+xI=yA`
`⇒[[16,8],[56,32]]+x[[1,0],[0,1]]=y[[3,1],[7,5]]`
`=>[[16+x,8+0],[56+0,32+x]=[[3y,y],[7y,5y]]`
`16+x=3y,y=8,7y=56,5y=32+x`
Putting `y=8` in `16+x=3y`, we get `x=24−16=8`
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