Introduction to non invertible matrix

Introduction to non invertible matrix:

Many people talk about matrices but do you know what exactly a matrix is? Let's see. A matrix is a set of numbers arranged in a particular order in a rectangular array with  m rows and n columns in it.Below is an example of matrix in which a_ij are know as elements or entities.

Matrices are of different types. They are

• Row Matrix
• Column Matrix
• Square Matrix
• Rectangular Matrix
• Diagonal Matrix
• Scalar Matrix
• Identity Matrix
• Null Matrix

Define Non Invertible Matrix:

A matrix is said to be non-invertible matrix if and only if its determinant is zero.Non-Invertible matrices are also known as Singular matrices.They are also known as Degenerate Matrices.

Conditions for the matrix to be non-invertible are

1. It should be a square matrix,i.e.the number of rows and columns in a matrix should be equal.
2. The determinant of square matrix should be zero.

Que : How to calculate the determinant of a matrix?

Ans : Step1 - Let  A be a 2 x 2 matrix

Step 2 - Multiply the elements in the first and second diagonals and subtract the product of the second diagonal from the first.

Formula for finding the value of the determinant = `a_(11)` *`a_(22)`  -  `a_(12)` *`a_(21)`

det A = 4 * 5 - 2 * 10

          = 20 - 20

         = 0

Example Problems Based on Non Invertible Matrix:

Ex : 1 Calculate the determinants for the following matrices and check if they are non-invertible matrices.

 `[[5,6],[2,15]]`

Sol : Step 1 - Let  A = `[[5,6],[2,15]]`

Step 2 - Formula = `a_(11)` *`a_(22)`  -  `a_(12)` *`a_(21)` 

det A = (5x15) -(2x6)

detA = 75 -12 = 63 . Since det A is not equal to zero, the given matrix is not a non invertible matrix.

Ex: 2 : Calculate the determinants for the following matrices and check if they are non-invertible matrices.

 `[ [1,0,0],[-2,0,0],[4,6,1]]`    

Sol :  Step 1 : Let A be the given matrix.

Formula : `a_(11)` *`a_(22)`  -  `a_(12)` *`a_(21)` 

Step 2: Plugging in the values in formula

det A = 1[(0 * 1) - (6 * 0)] - 0[(-2 * 1) - (4 * 0)] + 0[(-2 * 6) - (4 * 0)]

det A = 0 - 0 + 0

det A = 0

Since det A is equal to zero ,the given matrix is a non-invertible matrix.