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.

• 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

Articles related to Trace of a Matrix:

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

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.