Question

Question:All nonzero symmetric matrices are invertible.

Short answer

The given statement is false.

Step-by-step solution

Definition of a symmetric matrix.

Suppose A is a n × nmatrix.

Then the matrix is said to be symmetric matrix, ifA^T= A.

A matrix is said to be invertible, if the determinant of the matrix is non-zero.

Check whether the given statement is a true or false.

The given statement is all nonzero symmetric matrices are invertible.

For example.

Take,$A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right].$

Then,

$\begin{array}{c}{A}^T={\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]}^T\\ =\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\\ =A\end{array}$

So,A is a symmetric matrix, but it is not invertible, because$det\left(A\right)=0.$

This means,every symmetric matrix cannot be invertible.

Then the given statement is false.