How to Find Trace of Matrix – Properties and Example

Here you will learn how to find trace of matrix, its properties and what is orthogonal matrix with example.

Let’s begin –

Trace of Matrix

The sum of the elements of the square matrix A lying along the principal diagonal is called the trace of A i.e (tr(A)). 

Thus if  A = $[a_{ij}]_{n\times n}$,

then tr(A) = $\sum_{i=1}^{n}$ $a_{ii}$ = $a_{11}$ + $a_{22}$ + ……… + $a_{nn}$.

How to Find Trace of Matrix :

for example, for 3×3 matrix, if A = $\begin{bmatrix} 2 & 1 & -1 \\ 3 &  -2 & 5 \\ 1 & 5 &  3  \end{bmatrix}$ 

then, trace of A or tr(A) = 2 + (-2) + 3 = 3

for example, for 2×2 matrix, if A = $\begin{bmatrix} 2 & 1 \\ 3 &  4  \end{bmatrix}$ 

then, trace of A or tr(A) = 2 + 4 = 6

Properties of Trace of a Matrix

Let A = $[a_{ij}]_{n\times n}$ and B = $[b_{ij}]_{n\times n}$ and $\lambda$ be a scalar then

(i)  tr($\lambda A$)  = $\lambda$ tr(A) 

(ii) tr(A + B) = tr(A) + tr(B)

(iii) tr(AB) = tr(BA)

Orthogonal Matrix

A square matrix is said to be orthogonal matrix if 

 $AA^T$ = I (Identity matrix)

Note :  The determinant value of orthgonal matrix is 1 or -1.

Example : Show that the matrix A = $\begin{bmatrix} cosx & sinx \\ -sinx &  cosx  \end{bmatrix}$ is a orthogonal matrix.

Solution : We have,

A = $\begin{bmatrix} cosx & sinx \\ -sinx &  cosx  \end{bmatrix}$

$A^{T}$ = $\begin{bmatrix} cosx & -sinx \\ sinx &  cosx  \end{bmatrix}$

Now, we have to find $AA^T$ = $\begin{bmatrix} cosx & sinx \\ -sinx &  cosx  \end{bmatrix}$$\begin{bmatrix} cosx & -sinx \\ sinx &  cosx  \end{bmatrix}$

= $\begin{bmatrix} cos^2x + sin^2x & -cosx.sinx + sinx.cosx \\ -sinx.cosx + sinx.cosx &  cos^2x + sin^2x  \end{bmatrix}$

= $\begin{bmatrix} 1 & 0 \\ 0 & 1  \end{bmatrix}$ = I (Identity matrix)

$\implies$ $AA^T$ = I

Hence, it is an orthogonal matrix.