Trace of a Matrix Examples

Example 1: Prove that the trace of an identity matrix of order “3 × 3” is 3.

Solution:

Let us consider an identity matrix of order “3 × 3” to prove the trace of an identity matrix of order “3 × 3” is 3.

I₃ =  

We know that,

tr(A) = a11 + a22 + a33

tr(A) = 1 + 1 + 1 =3

Hence, proved.

Example 2: Calculate the trace of the matrix given below.

B = 

Solution:

From the given matrix,

b₁₁ = 1, b₂₂ = 11, b₃₃ = −5, and b₄₄ = −4.

We know that,

tr(A) = b₁₁ + b₂₂ + b₃₃ + b₄₄

= 1 + 11 + (−5) + (−4)

= 12 −5 −4 = 12 − 9 = 3

Thus, the trace of the given matrix B is 3.

Example 3: Calculate the trace of the matrix given below.

Solution:

From the given matrix,

a₁₁ = 0, a₂₂ = 24, a₃₃ = 7, a₄₄ = −5, and a₅₅ = 16.

We know that,

tr(A) = a₁₁ + a₂₂ + a₃₃ + a₄₄ + a₅₅

= 0 + 24 + 7 + (−5) + 16

= 47 −5 = 42

Thus, the trace of the given matrix A is 42.

Example 4: If R = P + Q, then prove that tr(R) = tr(P) + tr(Q), where “P, Q, and R” are square matrices of order “2 × 2”

Solution:

Let P = 

Q =  

R = P + Q

= 

Now, tr(R) = p₁₁ + q₁₁ + p₂₂ + q₂₂

tr(R) = p₁₁ + p₂₂ + q₁₁ + q₂₂

tr(P) = p₁₁ + p₂₂

tr(Q) = q₁₁ + q₂₂

tr(P) + tr(Q) = p₁₁ + p₂₂ + q₁₁ + q₂₂

tr(P) + tr(Q) = tr(R)

Hence, proved.

Trace of a Matrix: A matrix is defined as a rectangular array of numbers that are arranged in rows and columns. The size of a matrix can be determined by the number of rows and columns in it. A matrix is said to be an “m by n” matrix when it has “m” rows and “n” columns and is written as an “m × n” matrix. For example, if a matrix has three rows and four columns, then the order of the matrix is “3 × 4.” We have different types of matrices, such as rectangular, square, triangular, symmetric, etc.

In this article, we will learn about the Trace of a matrix, along with its definition, Trace of a Matrix properties, and Trace of a Matrix examples.