# 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.