The trace of a matrix is a fundamental concept in linear algebra, playing a critical role in various mathematical and applied disciplines. Defined as the sum of the diagonal elements of a square matrix, the trace provides significant insights into the properties of matrices and their applications.

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.

What is the 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.”

Trace of a Matrix Definition:

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 

Let us consider a square matrix of order “3 × 3,” as shown in the figure given below, a₁₁, a₁₂, a₁₃,…, a₃₂, and 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.

Trace of a Matrix Properties

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.

Linearity of the Trace

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) 

Trace of a Transpose

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

tr(A) = tr (A^T)

Trace of a Scalar Multiple

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

tr(kA) = k Tr(A)

Trace of a Product

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.

Trace of an Identity Matrix

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

tr(I_n) = n

Trace of a Zero Matrix

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

tr(O) = 0

Articles related to Trace of a Matrix:

• Minors and Cofactors of Determinants
• Adjoint of a Matrix
• Determinant of a Matrix

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 =  [Tex]\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right][/Tex]

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 = [Tex]\left[\begin{array}{cccc} 1 & -3 & 4 & 7\\ 2 & 11 & -9 & 6\\ 17 & 8 & -5 & 3\\ 5 & -22 & 14 & -4 \end{array}\right][/Tex]

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 = [Tex]\left[\begin{array}{cc} p_{11} & p_{12}\\ p_{21} & p_{22} \end{array}\right][/Tex]

Q =  [Tex]\left[\begin{array}{cc} q_{11} & q_{12}\\ q_{21} & q_{22} \end{array}\right][/Tex]

R = P + Q

= [Tex]\left[\begin{array}{cc} p_{11}+q_{11} & p_{12}+q_{12}\\ p_{21}+q_{21} & p_{22}+q_{22} \end{array}\right][/Tex]

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.

Practice Problems on Trace of a Matrix

1. Given the matrix [Tex]A = \begin{pmatrix} 3 & 5 \\ 1 & 7 \end{pmatrix} [/Tex], calculate the trace of matrix A.

2. Find the trace of the matrix [Tex]B = \begin{pmatrix} 4 & 1 & 7 \\ 0 & 6 & 8 \\ 2 & 3 & 5 \end{pmatrix} [/Tex]

3. Given the diagonal matrix [Tex]C = \begin{pmatrix} 9 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{pmatrix} [/Tex] compute the trace of matrix C

Trace of a Matrix – FAQs

What is 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.

What is meant by the trace of a matrix?

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.”

What is the trace of a null matrix?

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

What is the trace of an identity matrix of order n?

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

What is the definition of 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.”

What is the trace of a matrix?

The trace of a matrix, specifically a square matrix, is the sum of its diagonal elements. This property holds significance in various mathematical computations and applications​.

How do you calculate the trace of a matrix?

To calculate the trace of a matrix, you simply sum up all the diagonal entries from the upper-left corner to the bottom-right corner of the matrix​.

What properties does the trace of a matrix have?

The trace of a matrix has several important properties:

• It is invariant under cyclic permutations of matrix products.
• The trace of a matrix and its transpose are the same.
• It behaves linearly, meaning the trace of a sum of matrices equals the sum of their traces, and the trace of a scalar multiplied by a matrix equals the scalar times the trace of the matrix​.

Is the trace of a matrix affected by a change in basis?

No, the trace of a matrix is independent of the basis. This means it remains the same regardless of how the matrix is represented, as it is intrinsically linked to the eigenvalues of the matrix, which are basis-independent​.

Can the trace of a matrix be negative?

Yes, the trace of a matrix can be negative if the matrix has negative diagonal elements. The trace is simply the sum of these diagonal values, which can be positive, negative, or zero​.

How is the trace related to the eigenvalues of a matrix?

The trace of a matrix equals the sum of its eigenvalues. This relationship is often used to derive properties of matrices in various fields of study​.