Matrix Determinants
The determinant is a unique number associated with a square matrix. In this lesson, we show how to compute the determinant for any
square matrix, and we introduce notation for matrix determinants.
Notation for a Determinant
There are at least three ways to denote the determinant of a square matrix.
Denote the determinant by vertical lines around the matrix name; thus, the determinant of matrix A would be indicated by |A|.
Another approach is to enclose matrix elements within vertical straight lines, as shown below.
A11 A12 A13
|A| = A21 A22 A23
A31 A32 A33
And finally, some references refer to the deteriminant of A as Det A. Thus, |A| = Det A.
On this website, we will use the first option; that is, we will refer to the determinant of A as |A|.
How to Compute the Determinant of a 2 x 2 Matrix
Suppose A is a 2 x 2 matrix with elements Aij, as shown below.
A11 A12
A =
A21 A22
We compute the determinant of A according to the following formula.
|A| = ( A11 * A22 ) - ( A12 * A21 )
How to Compute the Determinant of an n x n Matrix
The formula for computing the determinant of a 2 x 2 matrix (shown above) is actually a special case of the general algorithm for
computing the determinant of any square matrix.
|A| = Σ ( + ) A1qA2rA3s . . . Anz
This algorithm requires some explanation. Here are the key points.
The determinant is the sum of product terms made up of elements from the matrix.
Each product term consists of n elements from the matrix.
Each product term includes one element from each row and one element from each column.
The number of product terms is equal to n! (where n! refers to n factorial).
By convention, the elements of each product term are arranged in ascending order of the left-hand (or row-designating)
The determinant is a unique number associated with a square matrix. In this lesson, we show how to compute the determinant for any
square matrix, and we introduce notation for matrix determinants.
Notation for a Determinant
There are at least three ways to denote the determinant of a square matrix.
Denote the determinant by vertical lines around the matrix name; thus, the determinant of matrix A would be indicated by |A|.
Another approach is to enclose matrix elements within vertical straight lines, as shown below.
A11 A12 A13
|A| = A21 A22 A23
A31 A32 A33
And finally, some references refer to the deteriminant of A as Det A. Thus, |A| = Det A.
On this website, we will use the first option; that is, we will refer to the determinant of A as |A|.
How to Compute the Determinant of a 2 x 2 Matrix
Suppose A is a 2 x 2 matrix with elements Aij, as shown below.
A11 A12
A =
A21 A22
We compute the determinant of A according to the following formula.
|A| = ( A11 * A22 ) - ( A12 * A21 )
How to Compute the Determinant of an n x n Matrix
The formula for computing the determinant of a 2 x 2 matrix (shown above) is actually a special case of the general algorithm for
computing the determinant of any square matrix.
|A| = Σ ( + ) A1qA2rA3s . . . Anz
This algorithm requires some explanation. Here are the key points.
The determinant is the sum of product terms made up of elements from the matrix.
Each product term consists of n elements from the matrix.
Each product term includes one element from each row and one element from each column.
The number of product terms is equal to n! (where n! refers to n factorial).
By convention, the elements of each product term are arranged in ascending order of the left-hand (or row-designating)
