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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` sot that `A^2+x I+y A=0` Hence, Find `A^(-1)`

Text Solution

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Given that, `A = [[3,1],[7,5]]`
`A^2 = [[3,1],[7,5]][[3,1],[7,5]]`
`Rightarrow A^2 = [[9+7,3+5],[21+35,7+25]] = [[16,8],[56,32]]`

Now, `A^2+xI+yA = 0`
`Rightarrow [[16,8],[56,32]]+x[[1,0],[0,1]]+y[[3,1],[7,5]] = 0`
`Rightarrow [[16+x+3y,8+y],[56+7y,32+x+5y]] = [[0,0],[0,0]]`
`Rightarrow 8+y = 0 => y = -8`
`Rightarrow 16+x+3y = 0 => 16+x+3(-8) = 0 => x = 8`
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