May 12, 2020

Introduction To Matrix Operations And Standard Determinants

A matrix is a rectangular array of numbers arranged in rows and columns. The representation of a matrix is as follows:

An m × n matrix consists of m rows which are horizontal and the n columns which are vertical.

Basic operations of a matrix involve matrix addition, scalar multiplication, transpose, matrix multiplication and row operations.

Operation	Definition	Example
Addition	The sum of two matrices A and B is denoted by A + B. It is calculated by adding each element of matrix A to the corresponding element of matrix B and is given by (A + B) = A + B.
Scalar multiplication	When a scalar C is multiplied to the matrix A, the product CA is obtained, which is called a scalar product. It is denoted by (CA). It is also called an inner product.
Transposition	The transpose of a matrix is obtained when the rows and columns are interchanged.

Matrix multiplication is a process of finding the product of two matrices A and B given that

the number of columns of the left matrix is the same as the number of rows of the right matrix. Suppose A is an m x n matrix and B is an n x p matrix, then the product matrix AB is m x p.

Row operations involve three types, namely:

Row addition: It deals with the addition of two rows.
Row multiplication: It deals with the multiplication of all the numbers of a row by a non-zero constant.
Switching of a row is interchanging two rows of a matrix.

The above matrix operations are done while solving linear equations and finding the inverse of a matrix.

Determinant of a matrix

The determinant det(A) or |A| of a square matrix A is a number that inscribes certain properties of the matrix. The necessary condition for a matrix to be invertible is that its determinant should be nonzero.

Consider a matrix A =

Then

Properties of determinants

The properties of the determinant include the following:

There will be a change in the sign if the rows and columns are interchanged.
The scalars can be factored out from the rows and columns.
There will be no change in the value of the determinant if the multiples of rows and columns are added together.
Suppose a row of a matrix is multiplied by a scalar c, then its determinant also gets multiplied by c.
The value of a determinant is 0 if a row or a column has zeros in it.
The value of a determinant is 0 if two rows or columns are equal.
If the elements of the row can be expressed as a sum of two or more elements, then the determinant can be expressed as the sum of two or more determinants.
The value of the determinant remains the same even if the equimultiples of corresponding elements of other rows or columns are added to each element of a row or a column of a determinant.
The value of the determinant is equal to the product of diagonal elements if all the elements of a determinant above or below the main diagonal consists of zeros.
x – α is a factor of ‘∆’ [value of determinant] if the value of ‘∆’ becomes zero when x = α is substituted.