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 aij 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
- It should be a square matrix,i.e.the number of rows and columns in a matrix should be equal.
- 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.
