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**

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**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 (R_{1}, R_{2} and R_{3}) and 3 columns (C_{1}, C_{2} and C_{3}).

- Expansion along first Row (R
_{1}) - Expansion along second Row (R
_{2}) - Expansion along third Row (R
_{3}) - Expansion along first Column (C
_{1}) - Expansion along second Column (C
_{2}) - Expansion along third Column (C
_{3})

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| = k^{n} | B |, where n = 1, 2, 3

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**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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