A square matrix A with elements form the set of real
numbers is said to be orthogonal if `A' = A^(-1).` If A is an
orthogonal matris, then

Options

A. A' is orthogonal
B. `A^(-1)` is orthogonl
C. `adj A = A'`
D. `abs(A^-1) =1`

Answer

A, B, D

Explanation

`because A' = A^(-1) rArr A A' = I ` ...(i)
Now, `(A' ) ' A' = I`
`therefore A'` is orthogonal
From Eq. (i) `(A A')^(-1) = I ^(-1) `
`rArr (A')^(-1) A^(-1) = I`
`rArr (A^(-1)'(A^(-1)) = I `
`therefore A^(-1) ` is orthogonal
Since, `adj A = A ^(-1) abs(A) ne A'`
and `abs(A^(-1)) = 1/abs(A) = pm 1` [ for orthogonal `abs(A) = pm 1`]