Trace of a Matrix

The sum of diagonal elements of a matrix is known as the trace of a matrix. The Trace of a matrix A can be represented as tr(A). The Trace of a matrix can be calculated for a square matrix only.

Example:

[Tex]A= \begin{bmatrix}  15& 12 &9 \\ 4& 6 &11  \\ 5& 9  &0   \\ \end{bmatrix}_{3\times 3}   [/Tex]

tr(A) = 15 + 6 + 0 = 21

Properties of a Trace of Matrix

i) Trace of the sum of two matrices is equal to the sum of a trace of an individual matrix.

Explanation:

Mathematically it can be written as tr(A+B) = tr(A) + tr(B)

[Tex]A= \begin{bmatrix}  15& 12 &9 \\ 4& 6 &11  \\ 5& 9  &0   \\ \end{bmatrix}_{3\times 3}   [/Tex]

tr(A) = 15 + 6 + 0 = 21 

[Tex]B= \begin{bmatrix}  4& 3 &7 \\ 8& 1 &2  \\ 5& 6  &1   \\ \end{bmatrix}_{3\times 3}   [/Tex]

tr(B) = 4 + 1 + 1 = 6

Now, tr(A)+tr(B)= 21+6 = 27

[Tex]A+B= \begin{bmatrix}  19& 15&16 \\ 12& 7 &13 \\ 10& 15  &1   \\ \end{bmatrix}_{3\times 3}   [/Tex]

tr(A + B) = 19 + 7 + 1 = 27

You can see, tr(A) + tr(B) = tr(A + tr(B)

Similarly, tr(A – B) = tr(A) – tr(B)

ii) Trace of a matrix that is multiplied by some scalar is equal to the multiplication of the trace of the matrix and scalar.

Explanation:

Mathematically it can be represented as tr(kA) = k tr(A)

[Tex]A=2\times \begin{bmatrix}  1& 4 &3    \\ 5& 7 &2  \\ 1& 3  &8   \\ \end{bmatrix}_{3\times 3}   [/Tex]

tr(2 × A) = 2 + 14 + 16 = 32

[Tex]2\times A= \begin{bmatrix}  2& 8 &6    \\ 10& 14 &4  \\ 2& 6  &16   \\ \end{bmatrix}_{3\times 3}   [/Tex]

2 × tr(A) = 2 * (1 + 7 + 8)

                = 32

You can see tr(2 × A) = 2 × tr(A)

Read More:

• Matrix Multiplication
• Adjoint of a Matrix
• Minors and Cofactors

Types of Matrices classify matrices into different categories based on the number of rows and columns present in them, the position of the elements, and also the special properties exhibited by the Matrix. A matrix is a rectangular array of numbers in which elements are arranged in rows and columns. Each element is identified as a_ij where i and j indicate the row and column number respectively for the element.

We have different types of matrices which are classified based on the number of rows and columns, the elements present in them, the order of the matrix, and the properties shown by the matrix. In this article, we will learn about different types of matrices in detail along with a brief introduction to matrices.