Class 12 Maths Determinants Order 1, 2 3

Determinant of a matrix of order one

Let A = [a] be the matrix of order 1, then determinant of A is defined to be equal to a

 

Determinant of a matrix of order two

 

 

Determinant of a matrix of order Three

 

 

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. There are 6 ways of expanding a determinant of order 3 corresponding to each of 3 rows (R1, R2 and R3) and 3 columns (C1, C2 and C3).

  • Expansion along first Row (R1)
  • Expansion along second Row (R2)
  • Expansion along third Row (R3)
  • Expansion along first Column (C1)
  • Expansion along second Column (C2)
  • Expansion along third Column (C3)

 

Refer ExamFear video lessons for more details.

 

  • For easier calculations, we shall expand the determinant along that row or column which contains maximum number of zeros.
  • While expanding, instead of multiplying by (–1)i + j, we can multiply by +1 or –1 according as (i + j) is even or odd

 

Numerical: Determine determinant of the matrix 

Solution: Determinant of this matrix =  (2 x -1) – (4 x -5)   = 18

 

Numerical: Determine determinant of the matrix 

Solution: Determinant of this matrix =   3 ( 0 x 0  -  (-5 x -1)) – (-1) (0 x 0 – (-1 x 3)) + (-2)( 0 x -5  - 0 x3)

= 3(-5) + 3 + 0

= -12   

 

 

Rule: If A = kB where A & B are square matrices of order n, then | A| = kn | B |, where n = 1, 2, 3

 

 

Numerical:

Solution:

det(2A) = |2A| = 2 x 4 -4 x 8  = 8 – 32  = -24

det(a) = |A| = 2 x 1 – 4 x 2  =-6

It can be seen that |2A| = 4 |A|

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