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Found in: Page 263

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# All nonzero symmetric matrices are invertible.

The given statement is false.

See the step by step solution

## Step 1: Definition of a symmetric matrix.

Suppose A is a${\mathrm{n}}{×}{\mathrm{n}}$ 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.

## Step 2: 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,

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

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.