Question

What is the “trace” of a Matrix \[?\]

Answer 1

Hint: To know the trace of a Matrix. First understand what a Matrix is. How the Matrix is defined, matrix properties and types of matrices. One such aspect is the trace of a matrix.

Complete step by step solution:
A matrix is a rectangular array of numbers (or other mathematical objects) arranged in rows and columns.
There are several types of matrices, but the most commonly used are:
1. Rows Matrix: A matrix is said to be a row matrix if it has only one row.
2. Columns Matrix: A matrix is said to be a column matrix if it has only one column.
3. Rectangular Matrix: A matrix is said to be rectangular if the number of rows is not equal to the number of columns.
4. Square Matrix: A matrix is said to be square if the number of rows is equal to the number of columns.
5. Diagonal Matrix: A square matrix is said to be diagonal if at least one element of the principal diagonal is non zero and all the other elements are zero.
6. Scalar Matrix: A diagonal matrix is said to be scalar if all of its diagonal elements are the same.
7. Identity Matrix: A diagonal matrix is said to be Identity if all of its diagonal elements are equal to one.
8. Transpose of a Matrix: Suppose \[A\] is a given matrix, then the matrix obtained by interchanging its rows into columns is called the transpose of \[A\]. it is denoted by \[{A^T}\].

Trace of a Matrix is the sum of the diagonal elements of the given square matrix. If the matrix is not square then the trace of a matrix does not exist.

Note: For any square matrix \[A\], The sum of the diagonal elements of\[A\] is equal to the sum of the eigenvalues of \[A\]. Then the trace of a matrix is the sum of the eigenvalues of the given matrix.