Question

If$\mathbf{A}$is a positive definite matrix, then the largest entry of$\mathbf{A}$must be on the diagonal.

Short answer

The given statement is FALSE.

Step-by-step solution

Step 1: Definition of positive definite matrix

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

Step 2: To Find TRUE or FALSE

Let $\mathrm{A}=\left[\begin{array}{cc}2& 70\\ 0& 1\end{array}\right]$.

Obviously, $\mathrm{A}$is an upper triangular matrix it’s eigen values and singular values are 1 and 2 and $\mathrm{A}$is thus positive definite.

The largest entry of $\mathrm{A}$however is not on the diagonal.

Step 3: Final Answer

The given statement is FALSE.