In the name of ALLAH the most beneficial and the most

merciful

SINGULAR & NON SINGULAR MATRICES

APPLIED LINEAR ALGEBRA – MATH 505

PRESENTER: SHAIKH TAUQEER AHMED STUDENT NUMBER# 433108347

SUBMITTED TO: DR. RIWZAN BUTT

PRESENTATION SCHEME

• Importance.• Definition.• Example of Singular Matrices.• Example of Non Singular Matrices.• Comparison

Importance

•By finding the given Matrix is Singular or Non-Singular we can determine weather the given system of linear equation has Unique Solution, No Solution or Infinitely Many Solutions.

DefinitionSingular Matrix:•If the determinant of a square matrix A is equal to zero then the matrix is said to be singular..

•The determinant is often used to find if a matrix is invertible . If the determinant of a square matrix is equal to zero, the matrix is not invertible, i.e., A-1 does not exist.

•For Example:

∴ Matrix A is Not invertible

014222412

AA

Example of Singular Matrix

• If one row of an n x n square matrix is filled entirely with zeros, the determinant of that matrix is equal to zero. • For Example:

004020012

AA

Example of Singular Matrix

• If two rows of a square matrix are equal or proportional to each other then the determinant of that matrix is equal to zero• Example of two rows equal:

• Example of two rows proportional:

012121212

AA

014222412

AA

Example of Singular Matrix

• A strictly upper triangular matrix is an upper triangular matrix having 0s along the diagonal as well as the lower portion.

• A strictly lower triangular matrix is a lower triangular matrix having 0s along the diagonal as well as the upper portion.

000

322300113120

nanaanaaa

U

021

023130012000

nana

aaa

L

Example of Singular Matrix• If any of the eigen values of A is zero, then A is singular

because

Det (A)=Product of Eigen Values

Let our nxn matrix be called A and let k stand for the eigen value. To find eigen values we solve the equation det(A-kI)=0

where I is the nxn identity matrix.

Assume that k=0 is an eigen value. Notice that if we plug zero into this

equation for k, we just get det(A)=0. This means the matrix is singluar

DefinitionNon-Singular Matrix:•If the determinant of a square matrix A is not equal to zero then the matrix is said to be Non-Singular..

•The determinant is often used to find if a matrix is invertible . If the determinant of a square matrix is not equal to zero, the matrix is invertible, i.e. A-1 exist.

•For Example:

∴ Matrix A is invertible

131592

9512

AA

EXAMPLE OF NON SINGULAR MATRIX

• A real symmetric matrix A is positive definite , if there exists a real non singular matrix such that

• A= M MT were MT is transpose

1101

,1011

,1001

EXAMPLE OF NON SINGULAR MATRIX

• A is called strictly diagonally dominant if

• For example

ij ijii AA

650153027

A

ComparisonNon Singular Singular

A is Invertible Non Invertible

Det(A) ≠0 =0

Ax=0 One solution x=0 Infinitely many solution

Ax=b One solution No solution or Infinitely many solution

A has Full rank r=n Rank r<n

Eigen Value All Eigen value are non-zero Zero is an Eigen value of A

AT A Is symmetric positive definite Is only semi-definite