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A matrix is said to be positive definite matrix, when it is symmetric matrix and all its eigenvalues are positive.

Let鈥檚 verify the given statement with the help of an example.

Consider a matrix $\mathrm{A}=\left[\begin{array}{cc}1& -1\\ -1& 3\end{array}\right]$

Find the symmetric of the matrix to prove that the given matrix is positive definite.

Because the symmetric of the matrix is said to be a positive definite matrix.

Now,

$$\begin{gathered} {\mathrm{A}}^{\mathrm{T}}=\left[\begin{array}{cc}1& -1\\ -1& 3\end{array}\right] \\ =\mathrm{A} \end{gathered}$$

Here, ${\mathrm{A}}^{\mathrm{T}}=\mathrm{A}$which means it鈥檚 symmetric. So, the given matrix is positive definite.

Also, the principle sub-matrices of $\mathrm{A}$have positive determinants.

Hence, $\mathrm{A}$is positive definite matrix because it is symmetric.

However,$\mathrm{A}$ have negative entries in it.

In view of this example, we can confirm that all the entries of a positive definite matrix are positive is FALSE.

So, the given statement is FALSE.