Evaluation of determinants of 2nd and 3rd Order Matrices

Definition:

Every square matrix can be associated to an expression or a number which is known as its determinant. If A = [a_ij] is a square matrix of order n then the determinant of A is denoted by  or |A| or 

Determinant of a square matrix of order 1:

If A = [a₁₁] is a square matrix of order 1, then the determinant of A is defined as |A| = a₁₁ or |a₁₁|= a₁₁

Determinant of a square matrix of order 2:

If is a matrix of order 2, then the expression a₁₁a₂₂ – a₁₂ a₂₁ is defined as the determinant of A i.e., 

The determinant of a square matrix of order 2 is equal to the product of the diagonal elements minus the product of the diagonal elements.

Determinant of square matrix of order 3:

If  is a square matrix of order 3, then the expression a₁₁a₂₂a₃₃ + a₁₂ a₂₃ a₃₁ + a₁₃ a₃₂ a₂₁ – a₁₁ a₂₃ a₃₂ – a₂₂ a₁₃ a₃₁ – a₁₂ a₂₁ a₃₃ is defined as the determinant of A.

Example: Evaluate determinant of 

Solution:

= 3 (-2 -1) + 2 (-1 -0) + 4 (1 - 0)

= -9 - 2 + 4

= - 7

• Only square matrices have determinants. The matrices which are not square do not have determinant.
• The determinant of a square matrix of order 3 can be expanded along any row or column.
• If a row or a column of a determinant consists of all zeros, then the value of the determinant is zero.
• The determinant of a skew symmetric matrix of odd order is zero.
  
       
• A determinant is called circulant if its rows (columns) are cyclic shifts of the first row (columns).
  

    It can be show that its value is 3abc – a₃ – b₃ – c₃