If A=[[3 ,1],[-1, 2]] , show that A^2-5A...

If `A=[[3 ,1],[-1, 2]]` , show that `A^2-5A+7I_2=O` .

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Given,
`A=[(3, 1),(-1 ,2)]`
`A^2=A*A`
`= [(3, 1),(-1 ,2)][(3, 1),(-1 ,2)]`
`=[(8, 5),(-5 ,3)]`
we know that `I=[(1,0),(0,1)]`
now,
`A^2-5A+7I_2`
`= [(8, 5),(-5 ,3)]-5[(3, 1),(-1 ,2)]+7[(1,0),(0,1)]`
`= [(8, 5),(-5 ,3)]-[(15, 5),(-5 ,10)]+[(7,0),(0,7)]`
`=[(0,0),(0,0)]`
Hence,`A^2-5A+7I_2=0`

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