Question

If A is an invertible symmetric matrix then
A. $(A^{-1})^{T}=A^{-1}$

B. adj $A=(adj~A)^{T}$

C. $A^{-1}$ is skew-symmetric

D. $|A|=0$

Choose the correct answer from the options given below

• A. A, B only
• B. A, C only
• C. B, C only
• D. C, D only

Hint:

Symmetry is preserved under inversion and adjoint operations. If a matrix starts symmetric, its inverse and adjoint will also be symmetric.

Answer 1

The Correct Option is A

We are given that A is an invertible symmetric matrix.
Invertible means $|A| \neq 0$, and symmetric means $A^T = A$.

Step 1: \color{redEvaluate Statement A
We know that for any invertible matrix, the operations of transpose and inverse commute: $(A^T)^{-1} = (A^{-1})^T$.
Since $A$ is symmetric, $A^T = A$.
Substituting this into the property: $A^{-1} = (A^{-1})^T$.
Therefore, the inverse of a symmetric matrix is also symmetric.
Statement A is correct.

Step 2: \color{redEvaluate Statement B
We know the relation $A^{-1} = \frac{1}{|A|} adj(A)$, which implies $adj(A) = |A| A^{-1}$.
Taking the transpose of both sides: $(adj~A)^T = (|A| A^{-1})^T = |A| (A^{-1})^T$.
From Step 1, we know $(A^{-1})^T = A^{-1}$.
So, $(adj~A)^T = |A| A^{-1} = adj~A$.
This confirms the adjoint of a symmetric matrix is symmetric.
Statement B is correct.

Step 3: \color{redEvaluate Statements C and D
As shown in Step 1, $A^{-1}$ is symmetric, not skew-symmetric.
Statement C is incorrect.
Invertible matrices must have a non-zero determinant ($|A| \neq 0$).
Statement D is incorrect.

Step 4: \color{redConclusion
Statements A and B are correct.
This matches Option (1).