2. SINGULAR & NON SINGULAR
MATRICES
APPLIED LINEAR ALGEBRA – MATH
505
PRESENTER: SHAIKH TAUQEER AHMED
STUDENT NUMBER# 433108347
SUBMITTED TO: DR. RIWZAN BUTT
4. 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.
5. Definition
Singular 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[ ] [ ][ ] [ ][ ] 01422
24
12
=−==
= AA
6. 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:
[ ] [ ][ ] [ ][ ] 00402
00
12
=−==
= AA
7. 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:
[ ] [ ][ ] [ ][ ] 01212
12
12
=−==
= AA
[ ] [ ][ ] [ ][ ] 01422
24
12
=−==
= AA
8. 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
3
22300
113120
na
naa
naaa
U
[ ]
=
021
02313
0012
000
nana
aa
a
L
9. 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
10. Definition
Non-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
95
12
=−==
= AA
11. 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
11
01
,
10
11
,
10
01
12. EXAMPLE OF NON SINGULAR MATRIX
• A is called strictly diagonally dominant if
• For example
∑ ≠
> ij ijii AA
[ ]
−
−
=
650
153
027
A
13. Comparison
Non 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
