Basic Determinant. A Topic of Maths.

2. What Really It Is ?

3. Determinant Of 2nd Order
Similarly :
Determinant Of 3rd Order
333
222
111
cba
cba
cba
22
11
ba
ba

4. In General :
Determinant Of nth Order
A block of n2 arranged in form of
n-rows and n-columns
Diagnol through left-hand top corner is
called Leading or Principal Diagnol
ln...
.
.
.
...
...
...
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.
.
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4...4444
3...3333
2...2222
1...1111
dncnbnan
ldcba
ldcba
ldcba
ldcba

5. • For 2nd Order determinant:
•
• For 3rd Order determinant:
• Co-factor is obtained by deleting the row &
column which intersect in that element with
proper sign.
• The sign of an element in the ith row and jth
column is (-1)i+j :The Co-factor of b3 i,e., B3 =
(-1)3+2 22
11
ca
ca
a1b2 – a2b122
11
ba
ba
333
222
111
cba
cba
cba
Co-Factors

6. Laplace’s Expansion
• A determinant can be expanded in terms of
any row( or column) as follows:












333
222
111
cba
cba
cba
Expanding by R1 (
i.e., 1st Row):
= a1A1 +b1B1 +c1C1
= a1
33
22
cb
cb
-b1 33
22
ca
ca
+
c1 33
22
ba
ba
Expanding by C2 ( i.e., 2nd
Column):
= -b1B1 +b2B2 –b3B3= -b1
33
22
ca
ca
+b2 33
11
ca
ca
-b3
22
11
ca
ca
1
2

7. Solution To is:
a1( b2c3 - b3c2) –b1( a2c3 – a3c2) + c1( a2b3 – a3b2)
1
Solution To is:
–b1( a2c3 – a3c2) +b2( a1c3 – a3c1) –b3( a1c2 – a2c1)
2

8. For Example: Find The Determinant Of
=
987
654
321
Expanding by R1 (
i.e., 1st Row):
= a1A1 +b1B1 +c1C1
= 1 98
65 -2
97
64 +3
87
54
= 1((5*9) – (6*8)) –2((4*9) – (6*7)) +3((4*8) – (5*7))
= 1(45 – 48) -2(36 – 42) +3(32 – 35)
= -3 +12 -3
=6

9. Properties Of Determinant :
1. A determinant remains unaltered by changing
its rows into columns and columns into rows.
2. If two parallel lines of a determinant are
interchanged, the determinant retains its
numerical value but changes in sign.
3. A determinant vanishes if two parallel lines are
identical.
4. If each element of a line be multiplied by same
factor, the whole detreminant is multiplied by
that factor.
5. If each element of a line consists of m terms,

10. Thank You