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Q7E

Expert-verifiedFound in: Page 263

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**All nonzero symmetric matrices are invertible.**

The given statement is false.

**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.**

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.

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