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3. Determinant of a 3x3 matrix: Determinant of a matrix of order three can be determined by expressing it in terms
of second order determinants. This is known as expansion of a determinant along a row (or a column).
a a a a a a
11 12 13 11 12 13
A = a a a A a a a
21 22 23 21 22 23
a a a a a a a
31 32 33 31 32 33
Expanding along Row 1
Following are the steps to calculate determinant of a matrix:
Step 1: Multiply first element a of Row1 by (–1) (1 + 1) [(–1)sum of suffixes in a ] and with the second order determinant
11
11
obtained by deleting the elements of first row and first column of | A | as a lies in Row1 and Column1.
11
a 11 a 12 a 13 a a
+
a a a ⇒ − ( 1) 11 a 22 23
21 22 23 11 a a
a 31 a 32 a 33 32 33
⇒ a (a × a – a × a )
23
33
11
32
22
nd
Step 2: Multiply 2 element a of Row1 by (–1)1 + 2 [(–1) sum of suffixes in a ] and the second-order determinant
12
12
obtained by deleting elements of the first row and 2 column of | A | as a lies in Row1 and Column2.
nd
12
a 11 a 12 a 13
(
+
1
a a a ⇒ − ) 12 a a 21 a 23
21 22 23 12 a a
a 31 a 32 a 33 31 33
⇒ –a (a × a – a × a )
12
33
31
23
21
Step 3: Multiply third element a of Row1 by (–1)1 + 3 [(–1) sum of suffixes in a 13 ] and the second-order determinant
13
obtained by deleting elements of the first row and third column of | A | as a lies in Row1 and Column3.
13
a 11 a 12 a 13 a a
a 21 a 22 a 23 ⇒ − 1 ) 13 a 13 a 21 a 22
+
(
a 31 a 32 a 33 31 32
⇒ –a (a × a – a × a )
21
13
32
22
31
Now the determinant of A, | A | is written as the sum of all three terms obtained in steps 1, 2 and 3 above:
So Finally combine all three –
= a (a × a – a × a ) – a (a × a – a × a )
A 11 22 33 32 23 12 21 33 31 23
+ a (a × a – a × a )
13
22
31
21
32
For example:
Calculate the determinant of the following matrix.
1 23
A = 4 05
7 80
Solution:
12 3
A = 40 5 Expanding alongRow1
78 0
05 45 40
= 1 − 2 + 3
80 70 78
166 Touchpad Artificial Intelligence (Ver. 2.0)-XI

