Learn Operations With Matrices

Matrices form an important tool in the study of finite-dimensional vector space. Determinants form an important tool in the study of matrices.

Introduction to Operations with Matrices:

Matrices are rectangular array items which are represented in rows and columns. Data in the matrices are called entry or an element. The matrix looks like

`[[7,2,1,3],[8,3,9,1]]`

The row of matrix is the horizontal line and column of matrix is the vertical line.

Operations on matrices are :

  • Addition
  • Subtraction
  • Multiplication
  • Transpose

Addition and subtraction of matrices

Addition of Matrices:

  •  The sum of two matrices should be of same order
  • Sum of two matrices is obtained by adding the corresponding entries.

Subtraction of Matrices:

  • The sum of two matrices should be of same order
  • Sum of two matrices is obtained by subtracting the corresponding entries.

Ex:  Let us consider A =  `[[3,2,1],[2,1,4],[1,5,3]]``[[2,3,1],[5,2,6],[3,7,1]]`

Sol:  Addition of matrices is done as shown below

Step I:         A + B =  `[[3+2,2+3,1+1],[2+5,1+2,4+6],[1+3,5+7,3+1]]`

Step II:                 =  `[[5,5,2],[7,3,10],[4,12,4]]`

Subtraction of matrices is done as shown below,

Step I:       A – B = `[[3-2,2-3,1-1],[2-5,1-2,4-6],[1-3,5-7,3-1]]`

Step II:                = `[[1,-1,0],[-3,-1,-2],[-2,-2,2]]`

Multiplication and transpose of Matrices:

Multiplication of matrics:

There are two types multiplication method done in matrices. They are

  • Scalar multiplication: Multiplication of a matrix by a scalar `alpha` is done by multilpying each entry by `alpha` .

 Ex: Let us consider the matrix A = `[[4,2,1],[3,5,2],[4,6,3]]`    and scalar `alpha = 2` 

Sol:   Scalar multiplication is calculated by

Step I:     `alpha` A = `[[2*4,2*2,2*1],[2*3,2*5,2*2],[2*4,2*6,2*3]]`

Step II:       2A = `[[8,4,2],[6,10,4],[8,12,6]]`

  • Matrix multiplication: Unless the number of column of first matrix and row of second matrix are similar, we cannot perform the matrix multiplication.

Ex:    Let us consider the matrix A = `[[2,3,4],[1,2,3],[4,1,2]]`        and        B = `[[1,2,3],[2,2,1],[1,2,3]]`   .

Sol:   Matrix multiplication is calculated by :

Step I:      A * B = `[[2+6+4,4+6+8,6+3+12],[1+4+3,2+4+6,3+2+9],[4+2+2,8+2+4,12+1+6]]`

Step II:               = `[[12,18,21],[8,12,14],[6,14,19]]`

Transpose:   In Transpose, the row and column of the matrix is interchanged to column and row respectively.

 Ex:    A = `[[1,2,3,4],[5,6,7,8]]` 

Sol:   Transpose of the matrix A is

Step I:         AT = `[[1,5],[2,6],[3,7],[4,8]]`