Determinants - July Dec Revision

12.1 Definition

Every square matrix can be associated to an expression or a number which is known as determinant.

If the matrix has only one element a₁₁ then a₁₁ is the determinant.

If the matrix is of order 2 that 2 by 2 matrix

|A| =

|a11 a12|
|a21 a22| =

a11*a22 – a12*a21

Determinant of a square matrix of order 3

Determinant of a square matrix of order 3 is the sum of the product of elements a1j in the first row with (-1) ^1+j times the determinant of a 2×2 sub-matrix obtained by leaving the first row and column passing through the element.



(i) Only square matrices have determinants.
(ii) The determinant of a square matrix of order three can be expanded along any row or column.

Determinant of a square matrix of order 4 or more

(iii) Determinant of a square matrix of order 4 or more can be determined following the procedure of finding the determinant of a square matrix of order 3. But in this case, especially in the case of 4×4 matrix, when we omit the rows and columns containing the elements of a row, we get 3×3 sub-matrices and we have to find determinants for them.

12.2 Singular matrix

A square matrix is a singular matrix if its determinant is zero.
Otherwise it is a non-singular matrix.


12.3 Minors and cofactors

Minor: For a square matrix [a_ij] or order n, the minor M_ij of a_ij, in A is the determinant of the square sub-matrix of order (n-1), obtained by leaving (or striking off) i^th row and j^th column of A.

Cofactor: Cofactor of an element a_ij in a square matrix [a_ij] is termed C_ij.

C_ij = (-1) ^i+j M_ij

M_ij is the minor of element a_ij in a square matrix [a_ij].

Minors and cofactors are defined for elements of a square matrix only. They are not defined for determinants.

12.4 Properties of determinants

1. For a square matrix, the sum of the product of elements of any row (or column) with their cofactors is always equal to determinants of the matrix.

2. For a square matrix, the sum of the product of elements of any row (or column) with the cofactors of corresponding elements of some other row (or column) is zero.

3. The value of a determinant remains unchanged if its rows and columns are interchanged.

4. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant changes by minus sign only.

5. If any two rows or columns of a determinant are identical then its value is zero.

6. If each element of a row (or a column) of a determinant is multiplied by a constant k, then the value of the new determinant is ‘k’ times the value of the original determinant.

7. If each element of a row (or a column) of a determinant is expressed as a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants of the same order.

8. If each element of a row (or a column) of a determinant is multiplied by the same constant and then added to the corresponding elements of some other row (column) then the value of the determinant remains same.

9. If each element of a row (or column) in a determinant is zero, then its value is zero.

10. If the matrix is a diagonal square matrix then its determinant is the product of all the diagonal elements.

11. If A and B are square matrices of the same order, then

|AB| = |A| |B|

12. If a matrix is a triangular matrix of order n, then its determinant is the product of all the diagonal elements.

12.5 Evaluation of Determinants

To evaluate determinants or large matrices, we use the properties of determinants given in section 12.4 above, to create many zeroes in the elements of a row or column and then expand the determinant using elements and cofactors of that row or column.

12.6 Evaluation of Determinants by using Factor Theorem

If f(x) is a polynomial and f(α) = 0 the, (x- α) is a factor of f(x).

If a determinant is a polynomial in x, then (x- α) is factor of the determinant if its value is zero when we put x = α.

Using this rule we can find determinant as a product of its factors.

12.7 Product of Determinants

A definition of product of determinants is similar to the rule of multiplication of matrices.




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