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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