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Friday, August 19, 2016

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 = [aij] 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

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