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| Determinants |
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| Let A = [aij] be a square matrix. We can associate with the square matrix A, a determinant which is formed by exactly the same array of elements of the matrix A. A determinant formed by the same array of elements of the square matrix A is called the determinant of the square matrix A and is denoted by the symbol det.A or |A|. |
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| It should be remembered that the determinant of a square matrix will be a scalar quantity. i.e., with a determinant we associate a definite value, whereas a matrix is essentially an arrangement of numbers and has no value. |
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| For example, |
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| To find the determinant of a matrix of order 3 (or more) we need the following definitions. |
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| Let |aij| be a determinant of order n. |
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| The determinant obtained by deleting the ith row and jth column in which the element aij lies is called the minor of element aij and is denoted by Mij. |
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| The co-factor of the element aij is (-1)i+j times its minor aij. We shall denote the cofactor of an element by the corresponding capital letter. |
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| Cofactor of aij = Aij = (-1)i+j Mij |
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| Consider the determinant |
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| The minor of a11 can be obtained by deleting the first row and first column of D. The Determinant, so obtained after deletion is the minor of a11 and is denoted by M11. |
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| The minor of a12 is obtained as follows |
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| Note that the position is 1st row and 2nd column. Delete the 1st row and 2nd column, the determinant so obtained is the minor of a12 |
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| That is minor of a12 |
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| The co-factor of a12 = A12 = (-1)1+2 M12 |
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| = - (a21a33 - a31a23) |
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| Similarly the minor of a13 |
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| = a21a32 - a31a22 |
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| Co-factor of a13 = A13 = (-1)1+3M13 |
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| = (a21a32 - a31a22) |
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| The value of the determinant D is given by a11A11 + a12A12 + a13A13 |
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| This is first row expansion of the determinant. Similarly, we can expand the determinant along any row or any column. |
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| In general, |
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| where i = 1, 2 and 3 |
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| or |
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| Where j = 1, 2 and 3. |
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 If the rows and columns of a determinant are inter-changed, the value remains unaltered. |
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If any two rows (columns) of a determinant are identical, its value of the determinant is zero. |
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If any two rows (columns) of a determinant are interchanged, the value of the determinant is (-1) times the original determinant. |
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If all the elements of one row (column) of a determinant is multiplied by k, the value of the new determinant is k times the original determinant. |
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If to any row or column of a determinant, a multiple of another row or column is added, the value of the determinant remains the same. |
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If some or all the elements of a row (or column) of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two or more determinants. Thus |
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The sum of the products of the elements of any row (column) with their corresponding cofactors is equal to the value of the determinant. |
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The sum of the products of the elements of any row (column) and the cofactors of the corresponding elements of any other row (column) is zero. |
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| Example: For a matrix of order 3, |
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| a11A21 + a12A22 + A13A23 = 0 |
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