Question

The inverse of a symmetric matrix (if it exists) is?

(a) A symmetric matrix

(b) A skew symmetric matrix

(c) A diagonal matrix

(d) A triangular matrix

The question was asked in an internship interview.

Query is from Finding Inverse and Rank of Matrix topic in chapter Matrices of Engineering Mathematics

Answer 1

Correct option is (a) A symmetric matrix

To explain: Let A be an invertible matrix.

We have AA^-1 = A^-1A = IN

(AA^-1)’ = (A^-1A)’ = (IN)’

(A^-1)’ A’ = A’ (A^-1)’ = IN

(A^-1)’ A = A (A^-1)’ = IN

(A^-1)’ = A^-1 (inverse of a matrix is unique).