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1.

For the matrix `A=[[3 ,1],[ 7, 5]],`find `x` and `y`sot that `A^2+x I+y Adot=0`Hence, Find `A^(-1)dot`

Answer» `A = [[3,1],[7,5]]`
`A^2 = [[3,1],[7,5]][[3,1],[7,5]]`
`=>A^2 = [[9+7,3+5],[21+35,7+25]] = [[16,8],[56,32]]`
Now, `A^2+xI+yA = 0`
`=>[[16,8],[56,32]]+x[[1,0],[0,1]]+y[[3,1],[7,5]] = 0`
`=>[[16+x+3y,8+y],[56+7y,32+x+5y]] = [[0,0],[0,0]]`
`=>8+y = 0 => y = -8`
`=> 16+x+3y = 0 => 16+x+3(-8) = 0 => x = 8`
`:. x = 8 and y = -8`
Now, `A^2+8I - 8A = 0`
`=>A^2-8A = -8I`
`=>-1/8[A^2-8A] = I`
`=>-1/8[A^2-8A]A^-1 = IA^-1`
`=>-1/8[A-8I] = A^-1`
`:. A^-1 = -1/8( [[3,1],[7,5]]-8[[1,0],[0,1]])`
`=>A^-1 = -1/8[[-5,1],[7,-3]]`
`A^-1 = [[5/8,-1/8],[-7/8,3/8]]`
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