Often, especially with determinants of large order, we can simplify the evaluation rules. In this Section we quote some useful properties of determinants in general.
If two rows (or two columns) of a determinant are interchanged then the value of the determinant is multiplied by
For example but (interchanging columns) and (interchanging rows)
The determinant of a matrix
and the determinant of its transpose
If two rows (or two columns) of a matrix
are equal then it has zero determinant.
For example, the following determinant has two identical rows:
If the elements of one row (or one column) of a determinant are multiplied by
, then the resulting determinant is
times the given determinant:
Note that if one row (or column) of a determinant is a multiple of another row (or column) then the value of the determinant is zero. (This follows from properties 3 and 4.)
This is predictable as the 3rd row is times the first row.
If we add (or subtract) a multiple of one row (or column) to another, the value of the determinant is unchanged.
Given , add (2 row 1) to (row 2) gives
The determinant of a lower triangular matrix, an upper triangular matrix or a diagonal matrix is the product of the elements on the leading diagonal.
As an example, it is easily confirmed that each of the following determinants has the same value .
This task is in four parts. Consider
Use property 2 to find another matrix whose determinant is equal to
, by transposing the matrix.
Now expand along the top row to express
as the sum of two products, each of a number and a
Use the statement after property 4 to show that the second of the
determinants is zero:
In the second determinant, row row 1 hence the determinant has value zero.
Use the statement after property 4 to evaluate the first determinant:
In the first determinant column 3 column 2. Hence this determinant is also zero. Therefore
Use Laplace expansion along the 1st row to determine
Show that the same value is obtained if you choose any other row or column for your expansion.
Using any of the properties of determinants to minimise the arithmetic, evaluate
Find the cofactors of
in the determinant
Prove that, no matter what the values of
Take out common factors in rows and columns
using then .
The value of the determinant (expand along top row) is then easily found
- Zero since (row 1) is (row 4).
- Take out common factors in rows and columns
- Cofactors of are respectively.