A = [β 8(π&π@π&π )]
[β 8(π& π@π& π )]
Find determinant of A = [β 8(3&2@1&4)]
|A| = 3 Γ 4 - 1 Γ 2
= 12 β 2
= 10
For a 3 Γ 3 matrix, like
A = [β 8(π&π&π@π &π&π@π&π&π)]
|β 8(π&π&π@π &π&π@π&π&π)|
=
|β 8( & @π&π@π&π)|
β
|β 8( & & @π & &π@π& &π)|
+
|β 8( & @π &π@π&π)|
|A| = a (ei β hf) β b (di β gf) + c (dh β eg)
Note : There is a + β pattern
+ β +
Letβs take an example
Find determinant of B = [β 8(9&2&3@5&β1&6@4&0&β2)]
|B| = 9 Γ |β 8(β1&6@0&β2)| β2 Γ |β 8(5&6@4&β2)| + 1 Γ |β 8(5&β1@4&0)|
= 9 ((β1) Γ (β2) β 0 Γ 6) β 2 (5 Γ (β2) β4 Γ 6) + 1 (5 Γ 0 β 4 Γ (β1))
= 9 (2 β0) β 2 (β10 β 24) + 1 (0 + 4)
= 9 Γ 2 β 2 Γ (β34) + 1 Γ 4
= 18 + 68 + 4
= 90
What about a 4 Γ 4 matrix?
For a 4 Γ 4 matrix, like
A = [β 8(π&π&π&π @π&π&π&π@π&π&π&π@π&π&π&π)]
Determinant is
|β 8( & & @π&π&π@π&π&π@π&π&π)|
|β 8( & & & @π& &π&π@π& &π&π@π& &π&π)|
|β 8( & & & @π& &π&π@π& &π&π@π& &π&π)|
|β 8( & & & @π&π& &π@π&π& &π@π&π& &π)|
|β 8( & & @π&π&π@π&π&π@π&π&π)|
Note : The + β pattern is followed
+ β + β
Matrix
Matrix is representation of number in row & column format
Eg: A = [β 8(9&2&1@5&β1&6@4&0&β2)]
Matrix can be of any order
[β 8(3@5@6)]_(3 Γ 1)
[β 8(3&2@1&4@5&3)]_(3 Γ 2)
[β 8(3&2@1&4)]_(2 Γ 2)
Scalar multiplied to matrix
If a number is multiplied to matrix, it is multiplied to each element of the matrix
2 [β 8(9&2&1@5&β1&6@4&0&β2)] = [β 8(2Γ9&2Γ2&2Γ1@2Γ5&2Γ(β1)&2Γ6@2Γ4&2Γ0&2Γ(β2))]
Determinant
Determinant is number associated with a matrix
Eg: |A| = |β 8(9&2&1@5&β1&6@4&0&β2)|
= 90
Determinant is only possible for a square matrix
|β 8(3&2@1&4@5&3)|
Determinant not possible
|β 8(3&2@1&4)|
Determinant possible
Scalar multiplied to determinant
If a number is multiplied to determinant, it is multiplied to either one row, or one column
2 |β 8(9&2&1@5&β1&6@4&0&β2)| = |β 8(2Γ9&2Γ2&2Γ1@5&β1&6@4&0&β2)|
Or
|β 8(2Γ9&2&1@2Γ5&β1&6@2Γ4&0&β2)|
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.