Matrix Operations

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:

matrix representation

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. 

AdditionThe 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 multiplicationWhen 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.
TranspositionThe 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. 

matrix A

Consider a matrix A =

determinant of matrix A


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.

Standard Determinants

The following are some of the standard determinants:

standard determinants