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.

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