Introduction To Matrix Operations And Standard Determinants - Sizling People
Home Education Introduction To Matrix Operations And Standard Determinants

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. 

OperationDefinitionExample
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

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.

Standard Determinants

The following are some of the standard determinants:

standard determinants

Must Read

Laser Lipo vs Coolsculpting: Which to choose?

a. Quick Facts Laser lipolysis and Cool sculpting, both are minimally invasive techniques of fat reduction. Laser Lipolysis uses...

Is It Time to Replace Your Water Line?

Residential main water lines are so often neglected. This is a shame since we are talking about one of the most important...

Tips On How to Get Dents Out of a Carpet

Carpet dents, also known as divots, are caused by heavy furniture compressing the fragile threads of your carpet for a lengthy period. They're...

Top 11 Diseases Your Face Will Reveal

Certain facial traits would reveal complicated health conditions. Most doctors predict it right by seeing your facial condition. Here in this blog,...

A Guide to Help You with Domain Name Search and Registration

With a domain name, you can represent yourself and your business on the Web. It is a unique identifier that helps your...
Download Best WordPress Themes Free Download
Download Premium WordPress Themes Free
Download Premium WordPress Themes Free
Download WordPress Themes Free
free download udemy paid course
download intex firmware
Download Nulled WordPress Themes
udemy course download free