The document discusses determinants and their properties. It defines determinants as representing single numbers obtained by multiplying and adding matrix elements in a special way. It then provides formulas for calculating determinants of matrices of order 1, 2 and 3. It also outlines several properties of determinants, such as how interchanging rows/columns, multiplying rows by constants, and adding rows affects the determinant. Finally, it discusses how determinants are used to determine whether systems of linear equations are consistent or inconsistent.

1. DETERMINANTS
• A Determinant of a matrix represents a single
number.
• We obtain this value by multiplying and
adding its elements in a special way.

2. Determinant of a Matrix of Order One
• Determinant of a matrix of order one
A=[a11]1x1 is
𝐴 = a11 = a11

3. Determinant of a Matrix of Order Two
• Determinant of a Matrix A=
𝑎11 𝑎12
𝑎21 𝑎22
2x2 is
𝐴 =
𝑎11 𝑎12
𝑎21 𝑎22
= 𝑎11 𝑎22 - 𝑎12 𝑎21

4. Determinant of a Matrix of Order Three
• Determinant of a Matrix A =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
is (expanding along R1)
• 𝐴 =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
= (-1)1+1 𝑎11
𝑎22 𝑎23
𝑎32 𝑎33
- (-1)1+2 𝑎12
𝑎21 𝑎23
𝑎31 𝑎33
+ (-1)1+3 𝑎13
𝑎21 𝑎22
𝑎31 𝑎32
= 𝑎11(𝑎22 𝑎33 - 𝑎32 𝑎23) - 𝑎12(𝑎21 𝑎33 - 𝑎31 𝑎23) + 𝑎13(𝑎21 𝑎32 - 𝑎31 𝑎22)
= 𝑎11 𝑎22 𝑎33 - 𝑎11 𝑎32 𝑎23 - 𝑎12 𝑎21 𝑎33 + 𝑎12 𝑎31 𝑎23 + 𝑎13 𝑎21 𝑎32 - 𝑎13 𝑎31 𝑎22

5. • http://www.authorstream.com/Presentation/j
oshsmith1110-162211-determinants-math-
ppt-shella-paglinawan-education-powerpoint/

6. Evaluate the Determinant ∆ =
1 2
4 2
•
1 2
4 2
= 1(2) – 2(4)
= 2 – 8
= -6

7. Find the Values of 𝑥 for which
3 𝑥
𝑥 1
=
3 2
4 1
•
3 𝑥
𝑥 1
=
3 2
4 1
• ie 3 – 𝑥2 = 3 - 8
⇒ 𝑥2 = 8
⇒ 𝑥 = ±2 2

8. Evaluate the Determinant ∆ =
3 −4 5
1 1 −2
2 3 1
•
3 −4 5
1 1 −2
2 3 1
= 3
1 −2
3 1
- (-4)
1 −2
2 1
+ 5
1 1
2 3
= 3(1 + 6) + 4(1 + 4) + 5(3 -2)
= 3(7) + 4(5) + 5(1)
= 21 + 20 + 5
= 46

9. Properties of Determinants
Property 1:The value of the determinant remains unchanged if its
rows and columns are interchanged.
Note: det(A) = det(A’) Where A’ = transpose of A

10. Property 2: If any two rows (or columns) of a determinant are
interchanged,then sign of determinant changes

11. Property 3: If any two rows (or columns) of a
determinant are identical (all corresponding elements
are same), then value of determinant is zero
• Verification: If we interchange the identical rows (or columns)
of the deteminent ∆, then ∆ does not change. But by Property
2 ∆ has changed its sign. Then ∆ = - ∆
ie ∆ = 0

12. Property 4: If each element of a row (or a column) of a
determinant is multiplied by a constant k, then its value
gets multiplied by k.

13. Note:
• Multiplying a determinant by k means
multiply elements of only one row (or one
column) by k.
• If A = [aij]3×3,then | k.A| = k3 |A|

14. Property 5:If some or all elements of a row or column of a determinant
are expressed as sum of two (or more) terms, then the determinant
can be expressed as sum of two (or more) determinants.

15. Property 6: To each element of any row or column of a determinant,
the equimultiples of corresponding elements of other row (or column)
are added, then the value of determinant remains the same

16. Area of a triangle
• Area of a triangle with vertices (x1, y1), (x2, y2)
and (x3, y3) is
∆ =
1
2
x1 y1 1
x2 y2 1
x3 y3 1

17. Minor
• Minor of an element aij of a determinant of
order n is the determinant of the square sub-
matrix of order (n-1) obtained by leaving ith
row and j th column.
• Minor of aij is denoted by Mij.

18. Find the minors of 1,-2,4,2 in
1 −2
4 2
• M11 = Minor of a11 = Minor of 1 = 2
• M12 = Minor of a12 = Minor of -2 = 4
• M21 = Minor of a21 = Minor of 4 = -2
• M22 = Minor of a22 = Minor of 2 = 1

19. Find the minors of all elements in
3 −4 5
1 6 −2
2 3 0
• M11 = Minor of a11 = Minor of 3 =
6 −2
3 0
= (6x0) – (-2x3) = 0 + 6 = 6
• M12 = Minor of a12 = Minor of -4=
1 −2
2 0
= (1x0) – (-2x2) = 0 + 4 = 4
• M13 = Minor of a13 = Minor of 5 =
1 6
2 3
= (1x3) – (2x6) = 3 -12 = -9
• M21 = Minor of a21 = Minor of 1 =
−4 5
3 0
= (-4x0) – (3x5) = 0 -15 = -15
• M22 = Minor of a22 = Minor of 6 =
3 5
2 0
= (3x0) – (2x5) = 0 - 10 = -10
• M23 = Minor of a23 = Minor of -2=
3 −4
2 3
= (3x3) – (-4x2) = 9+8 = 17
• M31 = Minor of a31 = Minor of 2 =
−4 5
6 −2
= (-4x-2)-(6x5) =8-30 =-22
• M32 = Minor of a32 = Minor of 3 =
3 5
1 −2
= (3x-2) – (1x5) = -6-5 =-11
• M33 = Minor of a33 = Minor of 0 =
3 −4
1 6
= (3x6) – (1x-4) = 18 +4 =22

20. Co factor
• Cofactor of an element aij , denoted by Aij is
Aij = (–1)i + j Mij
where Mij is minor of aij.
Note: Aij =
𝑀𝑖𝑗 𝑖𝑓 𝑖 + 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛
−𝑀𝑖𝑗 𝑖𝑓 𝑖 + 𝑗 𝑖𝑠 𝑜𝑑𝑑
If elements of a row (or column) are multiplied with cofactors of any other row
(or column), then their sum is zero.

21. Find the cofactors of 1,-2,4,2 in
1 −2
4 2
• M11 = Minor of a11 = Minor of 1 = 2
A11= Cofactor of a11 = (-1)1+1M11 =2
M12 = Minor of a12 = Minor of -2 = 4
A12= Cofactor of a12 = (-1)1+2M12 = -4
• M21 = Minor of a21 = Minor of 4 = -2
A21= Cofactor of a21 = (-1)2+1M21 = 2
• M22 = Minor of a22 = Minor of 2 = 1
A22= Cofactor of a22 = (-1)2+2M22 = 1

22. Adjoint of a matrix
• Adjoint of a matrix A = [aij]n × n is the transpose of the
matrix [Aij]n × n, where Aij is the cofactor of the element aij .
Denoted by adj A.
• If A =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
adj A = Transpose of
𝐴11 𝐴12 𝐴13
𝐴21 𝐴22 𝐴23
𝐴31 𝐴32 𝐴33
=
𝐴11 𝐴21 𝐴31
𝐴12 𝐴22 𝐴32
𝐴13 𝐴23 𝐴33

23. Theorem 1 : If A be any square matrix of order n, then
A(adj A) = |A|In = (adj A) A
Verification: Let A =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
then adj A =
𝐴11 𝐴21 𝐴31
𝐴12 𝐴22 𝐴32
𝐴13 𝐴23 𝐴33
Sum of the product of elements of a row (or a column) with corresponding
cofactor is equal to lAl, otherwise zero.
∴ A (adj A) =
lAl 0 0
0 lAl 0
0 0 lAl
= lAl
1 0 0
0 1 0
0 0 1
= lAl I ..…. (i)
Similarly (adj A) A = lAl I …… (ii)
By (i) & (ii) A(adj A) = |A|I n = (adj A) A

24. SINGULAR & NON SINGULAR
• Singular:
A square matrix A is said to be singular if
| A | = 0
• Non Singular
A square matrix A is said to be non-singular if
|A| ≠ 0.

25. Theorem 2: If A and B are nonsingular matrices of the
same order, then AB and BA are also nonsingular
matrices of the same order.
i.e. If |A|≠ 0 & |B|≠ 0
Then |AB|≠ 0 & |BA|≠ 0
Theorem 3: The determinant of the product of matrices is
equal to the product of their respective determinants.
i.e. |AB| =|A| |B|

26. Invertible Matrices
• If A & B are Square Matrices such that
AB = BA = I
B is called inverse matrix of A
B = A-1
A is said to be invertible

27. Theorem 4 : A square matrix A is invertible if and only if A is
nonsingular matrix
Verification: Let A be an invertible matrix. Then there exists a matrix
B such that AB = In = BA
⇒ |AB| =| In |
⇒ |A| |B| = I [ ∵ |AB| =|A| |B|]
⇒ |A| ≠ 0
⇒ A is a non-singular matrix
Conversly, let A be a non-singular matrix of order n. Then |A| ≠ 0
A(adj A) = |A|In = (adj A) A (by thm 1)
⇒ A
1
|A|
adj A = In =
1
|A|
adj A A
⇒ A-1 =
1
|A|
adj A
Hence A is an invertible matrix.

28. Consistency of the System of Linear Equation
Consistent system
A system of equations is said to be consistent
if its solution (one or more) exists.
Inconsistent system
A system of equations is said to be
inconsistent if its solution does not exist.

29. Solution of system of linear equations using inverse of
a matrix
If a1x+b1y+c1z = d1
a2x+b2y+c2z = d2
a3x+b3y+c3z = d3
writing these equation as AX = B
where A =
𝑎1 𝑏1 𝑐1
𝑎2 𝑏2 𝑐2
𝑎3 𝑏3 𝑐3
, X =
𝑥
𝑦
𝑧
and B =
𝑑1
𝑑2
𝑑3
Then X = A-1B
(i) |A|≠ 0 , there exists unique solution.
(ii) |A| = 0 and (adj A)B ≠ 0, then there exists no solution.
(iii)|A| = 0 and (adj A)B = 0, then system may or may not be consistent.