If A is a square matrix of order less than 4 such that `|A-A^(T)| ne 0` and `B=` adj. (A), then adj. `(B^(2)A^(-1) B^(-1) A)` is

Question

If A is a square matrix of order less than 4 such that `|A-A^(T)| ne 0` and `B=` adj. (A), then adj. `(B^(2)A^(-1) B^(-1) A)` is

If A is a square matrix of order less than 4 such that `|A-A^(T)| ne 0` and `B=` adj. (A), then adj. `(B^(2)A^(-1) B^(-1) A)` is
A. `A`
B. `B`
C. `|A|A`
D. `|B|B`

Monjur Alahi

6 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - A
We know that `A-A^(T)` is skew-symmetric.
Now, it is given that `|A-A^(T)| ne 0`.
So, order of A cannot be odd.
Thus, order of A is 2.
Now, `B=` adj. A
`:. AB=BA=|A|I`
`implies (AB)^(-1)=(BA)^(-1)`
`implies B^(-1) A^(-1)=A^(-1) B^(-1)`
`:.` adj. `(B^(2) A^(-1) B^(-1) A)=` adj. `(B^(2) B^(-1) A^(-1) A)`
= adj. B = adj. (adj. A)`=|A|^(2-2) A=A`