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.

Square Matrix

Before learning the concept of the trace of a matrix, we need to know about a square matrix. A square matrix is defined as a matrix that has an equal number of rows and columns. For example, if the order of a square matrix is “3 × 3,” then it has three rows and three columns.

Square Matrix of order “2 × 2”

A = 

Square matrix of order “3 × 3”

B = 

What is Trace of a Matrix?

Trace of a matrix is defined as the sum of the principal diagonal elements of a square matrix. It is usually represented as tr(A), where A is any square matrix of order “n × n.” Let us consider a square matrix of order “3 × 3,” as shown in the figure given below, a₁₁, a₁₂, a₁₃,…, a₃₂, a₃₃ are the entries of the given matrix A. Now, the trace of matrix “A” is equal to the sum of its principal diagonal elements, i.e., a₁₁, a₂₂, and a₃₃.

If A is a square matrix of order “n × n,” then the trace of matrix A is equal to the sum of the main diagonal elements.

tr(A) = a₁₁ + a₂₂ + a₃₃ + …+ a_nn

Properties of Trace of a Matrix

The following are some important properties of a trace of a matrix. Let us consider two square matrices A and B of the same order.

• The sum of the traces of the matrix A and the matrix B is equal to the trace of the matrix that is obtained by the sum of the matrices A and B.

tr(A) + tr(B) = tr (A + B) 

• The trace of a given matrix and its transpose are the same.

tr(A) = tr (A^T)

• If A is any square matrix of order “n × n” and k is a scalar, then

tr(kA) = k Tr(A)

• If A is a matrix of order “m × n” and B is a matrix of order “n × m,” then the trace of AB is equal to the trace of BA.

tr (AB) = tr (BA)

The above statement is true if both AB and BA are defined.

•  The trace of an identity matrix of order “n × n” is n.

tr(I_n) = n

• The trace of a zero or null matrix of any order is zero.

tr(O) = 0

Solved Examples on Trace of a Matrix

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.

FAQs on Trace of a Matrix

Question 1: What is a square matrix?

Answer:

A square matrix is defined as a matrix that has an equal number of rows and columns. For example, if the order of a square matrix is “3 × 3,” then it has three rows and three columns.

Question 2: What is meant by the trace of a matrix?

Answer:

The trace of a matrix is defined as the sum of the principal diagonal elements of a square matrix. It is usually represented as tr(A), where A is any square matrix of order “n × n.”

Question 3: What is the trace of a null matrix?

Answer:

The trace of a zero or null matrix of any order is zero, i.e., tr(O) = 0.

Question 4: What is the trace of an identity matrix of order n?

Answer:

The trace of an identity matrix of order “n × n” is n, i.e., tr (I_n) = n.

tr (I_n) = 1 + 1+ 1 + …+ 1 (n times) = n