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$\mathrm{n}\times \mathrm{n}$matrix.

Then the matrix is said to be symmetric matrix, if${\mathrm{A}}^{\mathrm{T}}=\mathrm{A}$.

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,

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

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

This means,every symmetric matrix cannot be invertible.

Then the given statement is false.