Question

Let A elementof M_n(R) be a diagonalizable matrix with tr(A^2) = 0. Prove that A is the zero matrix.

Answer 1

Answer with Step-by-step explanation:

Let A element of $M_n$(R) be a diagonalizable matrix with $tr(A^2)=0$ given

We have to prove that A is the zero matrix

Le A=$\left[\begin{array} {ccc}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\end{array}\right]$  be any  matrix of order$3\times 3$

Trace : Trace is defined as the sum of diagonal elements of a matrix.

$A\times A=A^2=\left[\begin{array}{ccc}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\end{array}\right]\times \left[\begin{array}{ccc}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\end{array}\right]$

$A^2=\left[\begin{array} {ccc}a^2_{11}&0&0\\0&a^2_{22}&0\\0&0&a^2_{33}\end{array}\right]$

Therefore, $tr(a^2_{11}+a^2_{22}+a^2_{33})=0$

We know that  square of an positive or negative element is positive

Therefore, it is necessary that $a_{11}=0,a_{22}=0,a_{33}=0$ when tr($A^2)$=0

When all elements of matrix are zero then the matrix should be zero matrix.

Hence, A is a zero matrix .