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Let A be an nth-order square matrix and ...

Let `A` be an nth-order square matrix and `B` be its adjoint, then `|A B+K I_n|` is (where `K` is a scalar quantity) a. `(|A|+K)^(n-2)` b. `(|A|+K)^n` c. `(|A|+K)^(n-1)` d. none of these

A

`(abs(A) +k)^(n-2) `

B

`(abs(A) +k)^(n)`

C

`(abs(A) +k)^(n-1)`

D

`(abs(A) +k)^(n+1)`

Text Solution

Verified by Experts

The correct Answer is:
B
` because ` B = adj A
`rArr AB = A("ajd " A) = abs(A) I_(n)`
`therefore AB + KI_(n )= abs(A) I_(n) + kI_(n) = (abs(A) + k ) I_(n)`
`rArr abs( AB + KI_(n ))= abs((abs(A) + k))I_(n) = (abs(A) + k )^(n)`
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