# Properties of Determinants

- The value of the determinant is not changed when rows are changed into corresponding columns.
- If any two rows or columns of a determinant are interchanged, the sign of the determinant is changed, but its magnitude remains the same.
- The value of a determinant is zero if any two rows of columns are identical.
- 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.
- 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,
- 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.
- If Î”
=_{r}*f*_{1}(*r*),*f*_{2}(*r*), and*f*_{3}(*r*) are functions of*r*and*a*,*b*,*c*,*d*,*e*, and*f*are constants. Then*x*) =*f*_{1}(*x*),*f*_{2}(*x*), and*f*_{3}(*x*) are functions of*x*and*a*,*b*,*c*,*d*,*e*, and*f*are constants. We have

# Some important determinants

- = (
*x*â€“*y*) (*y*â€“*z*) (*z*â€“*x*) - = (
*x*â€“*y*) (*y*â€“*z*) (*z*â€“*x*) (*x*+*y*+*z*) - = (
*x*â€“*y*) (*y*â€“*z*) (*z*â€“*x*) (*xy*+*yz*+*zx*).

# Product of two determinants

Let Î”

_{1}= and Î”_{2}=Then row by row multiplication of Î”

_{1}and Î”_{2}is given byÎ”

_{1}Ã— Î”_{2}=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

*A*

_{1},

*B*

_{1},

*C*

_{1}, â€¦, are respectively the cofactors of the elements

*a*

_{1},

*b*

_{1},

*c*

_{1}, â€¦, of the determinant

Î” = , Î” â‰ 0, then = Î”

^{2}# Differentiation of a determinant

- Let Î”(
*x*) be a determinant of order 2. If we write Î”(*x*) = [*C*_{1};*C*_{2}], where*C*_{1}and*C*_{2}denote the first and second columns then Î”â€²(*x*) = [*C*â€²_{1};*C*_{2}] + [*C*_{1};*C*â€²_{2}], where*C*â€²denotes the column which contains the derivative of all the functions in the_{i}*i*th column*C*. In a similar fashion, if we write_{i}*x*) = , then Î”â€² (*x*) =*x*) = ,*x*> 0, then*x*) = - Let Î”(
*x*) be of order 3. If we write Î”(*x*) = [*C*_{1};*C*_{2};*C*_{3}], then Î”â€²(*x*) = [*C*â€²_{1};_{ }*C*_{2};*C*_{3}] + [*C*_{1};*C*â€²_{2};*C*_{3}] + [*C*_{1};*C*_{2};*C*â€²_{3}] and similarly if we consider*x*) = , then Î”â€²(*x*) = - If only one row (column) consists functions of
*x*and other rows are constants, viz., let*x*) =*x*) =(^{n}*x*) =*n*is any positive integer and*f*^{n}(*x*) denotes the*n*th derivative of*f*(*x*).