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Let A be an orthogonal square matrix. ...

Let A be an orthogonal square matrix.
Statement -1 : `A^(-1)` is an orthogonal matrix.
Statement -2 : `(A^(-1))^T=(A^T)^(-1) and (AB)^(-1)=B^(-1)A^(-1)`

A

Statement -1 is True, Statement -2 is true, Statement -2 is a correct explanation for Statement-1.

B

Statement-1 is True, Statement -2 is True, Statement -2 is not a correct explanation for Statement -1.

C

Statement -1 is True, Statement -2 is False.

D

Statement -1 is False, Statement -2 is True.

Text Solution

Verified by Experts

The correct Answer is:
A
Clearly, statement -2 is ture (see theorems 4 on page 16,7). Since A is an orthogonal matrix.
`:. A A^T=A^TA=I`
`rArr (A A^T)^(-1)=(A^TA)^(-1)=I`
`rArr (A^T)^(-1)A^(-1)=I=A^(-1)(A^T)^(-1) [:' (AB)^(-1)=B^(-1)A^(-1)]`
`rArr (A^(-1))^TA^(-1)=A^(-1)(A^(-1)^T [:' (A^T)^(-1)=(A^(-1))^T]`
`A^(-1)` is orthogonal.
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Knowledge Check

  • Let A be a square matrix of order n. Statement - 1 : abs(adj(adj A))=absA^(n-1)^2 Statement -2 : adj(adj A)=absA^(n-2)A

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