Answer with Step-by-step explanation:
Let A element of
(R) be a diagonalizable matrix with
given
We have to prove that A is the zero matrix
Le A=
be any matrix of order

Trace : Trace is defined as the sum of diagonal elements of a matrix.
![A* A=A^2=\left[\begin{array}{ccc}a_(11)&0&0\\0&a_(22)&0\\0&0&a_(33)\end{array}\right]* \left[\begin{array}{ccc}a_(11)&0&0\\0&a_(22)&0\\0&0&a_(33)\end{array}\right]](https://img.qammunity.org/2020/formulas/mathematics/college/habvpotyem65wmiwmwbsspt4p2zakf0b24.png)
Therefore,

We know that square of an positive or negative element is positive
Therefore, it is necessary that
when tr(
=0
When all elements of matrix are zero then the matrix should be zero matrix.
Hence, A is a zero matrix .