Trace of a Matrix (original) (raw)

Last Updated : 13 Mar, 2026

The trace of a matrix refers to the sum of the diagonal elements in a square matrix. It is a key concept in linear algebra and is widely used in mathematics, physics, machine learning, and other applied fields.

For a square matrix A of order n×n, the trace is denoted as tr(A) or trace(A) and is defined as the sum of the principal diagonal elements:

**tr(A) = a 11 **+ a 22 **+ a 33 **+ ⋯ + a nn or **trace(A) = ⅀ n n=1 A nn

where a11, a22, a33, …,​ are the diagonal elements of the matrix A

**For example, 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.

Trace of a Matrix

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

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

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

**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.

Example 3

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

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

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

**Question 1: Given the matrixA = \begin{pmatrix} 3 & 5 \\ 1 & 7 \end{pmatrix}, calculate the trace of matrix A.

**Question 2: Find the trace of the matrix B = \begin{pmatrix} 4 & 1 & 7 \\ 0 & 6 & 8 \\ 2 & 3 & 5 \end{pmatrix}

**Question 3: Given the diagonal matrix, C = \begin{pmatrix} 9 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{pmatrix}compute the trace of matrix C.

**Question 4: If the trace of matrix A = = \begin{pmatrix} 2+x & 3 & 4 \\ 1 & -1 & 2 \\ -5 & 1 & x \end{pmatrix} is 5, then find the value of x.

**Answer key

**Answer 1: The trace of the matrix equals **10.
**Answer 2: The trace of the matrix equals 15.
**Answer 3: The trace of the matrix equals **17.
**Answer 4: The value of x is 2