Question

Suppose a matrix A satisfies `A^2-5A+7I=0`. If `A^5=aA+bI`, then the value of `2a+b` is:

Answer 1

`A^2 - 5A + 7I = 0`
`A^2 = 5A-7I`
`=>A^2*A = A(5A-7I)`
`=>A^3 = 5A^2-7A*I`
`=>A^3 = 5(5A-7I) - 7A`
`=>A^3 = 25A - 35I-7A`
`=>A^3 = 18A-35I`
`=>A^3*A = A(18A-35I)`
`=>A^4 = 18A^2-35A*I`
`=>A^4 = 18(5A-7I) - 35A`
`=>A^4 = 90A - 126I-35A`
`=>A^4 = 55A-126I`
`=>A^4*A = A(55A-126I)`
`=>A^5 = 55A^2-126A*I`
`=>A^5 = 55(5A-7I) - 126A`
`=>A^5 = 275A - 385I-126A`
`=>A^5 = 149A-385I->(1)`
Now, it is given that , `A^5 = aA+bI`
Comparing it with (1),
`a = 149, b = -385`
`:. 2a+b = 2(149) - 385 = -87.`