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Properties of Determinants

  1. The value of the determinant is not changed when rows are changed into corresponding columns.
    Naturally when rows are changed into corresponding columns, then columns will be changed into corresponding rows. That is,
  2. If any two rows or columns of a determinant are interchanged, the sign of the determinant is changed, but its magnitude remains the same.
  3. The value of a determinant is zero if any two rows of columns are identical.
  4. A common factor of all elements of any row (or of any column) may be taken outside the sign of the determinant. In other words, if all the elements of the same row (or the same column) are multiplied by a certain number, then the determinant gets multiplied by that number.
    i.e., 65030.png
    [taking 8 common from the first row]
    = 65024.png
    [taking 4 common from the first column]
  5. If every element of some column or (row) is the sum of two terms, then the determinant is equal to the sum of two determinants; one containing only the first term in place of each sum, the other only the second term. The remaining elements of both determinants are the same as in the given determinant. That is,
  6. The value of a determinant does not change when any row or column is multiplied by a number or an expression and is then added to or subtracted from any other row or column.
    Here it should be noted that if the row or column which is changed is multiplied by a number, then the determinant will have to be divided by that number. That is,
  7. If Δr = 65006.png
    where f1(r), f2(r), and f3(r) are functions of r and a, b, c, d, e, and f are constants. Then
    Also for
    Δ(x) = 64994.png
    where f1(x), f2(x), and f3(x) are functions of x and a, b, c, d, e, and f are constants. We have

Some important determinants

  1. 64982.png= (xy) (yz) (zx)
  2. 64976.png= (xy) (yz) (zx) (x + y + z)
  3. 64970.png= (xy) (yz) (zx) (xy + yz + zx).

Product of two determinants

Let Δ1 = 64963.png and Δ2 = 64957.png
Then row by row multiplication of Δ1 and Δ2 is given by
Δ1 × Δ2 = 64950.png
Multiplication can also be performed row by column; column by row or column by column as required in the problem.
To express a determinant as product of two determinants, one requires a lots of practice and this can be done only by inspection and trial.
Property If A1, B1, C1, …, are respectively the cofactors of the elements a1, b1, c1, …, of the determinant
Δ = 64943.png, Δ 0, then 64937.png = Δ2

Differentiation of a determinant

  1. Let Δ(x) be a determinant of order 2. If we write Δ(x) = [C1; C2], where C1 and C2 denote the first and second columns then Δ′(x) = [C1 ; C2] + [C1; C2], where Ci denotes the column which contains the derivative of all the functions in the ith column Ci. In a similar fashion, if we write
    Δ(x) = 64931.png, then Δ′ (x) = 64925.png
    For example, let Δ(x) = 64919.png, x > 0, then
    Δ′(x) = 64913.png
  2. Let Δ(x) be of order 3. If we write Δ(x) = [C1; C2; C3], then Δ′(x) = [C1; C2; C3] + [C1; C2; C3] + [C1; C2; C3] and similarly if we consider
    Δ(x) = 64907.png, then Δ′(x) = 64901.png
  3. If only one row (column) consists functions of x and other rows are constants, viz., let
    Δ(x) = 64895.png
    Δ′(x) = 64887.png
    and in general
    Δn(x) = 64881.png
    where n is any positive integer and fn (x) denotes the nth derivative of f(x).

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