Let A be an n×n matrix. The determinant of A, denoted as det
Definition 3.1.1 Let A =

The determinant is also often denoted by enclosing the matrix with two vertical lines. Thus det
Example 3.1.2 Find det
From the definition this is just
Assuming the determinant has been defined for k ×k matrices for k ≤ n− 1, it is now time to define it for n × n matrices.
Definition 3.1.3 Let A =
Now here is the definition of the determinant given recursively.
Theorem 3.1.4 Let A be an n × n matrix where n ≥ 2. Then
 (3.1) 
The first formula consists of expanding the determinant along the i^{th} row and the second expands the determinant along the j^{th} column.
Note that for a n×n matrix, you will need n! terms to evaluate the determinant in this way. If n = 10, this is 10! = 3,628,800 terms. This is a lot of terms.
In addition to the difficulties just discussed, why is the determinant well defined? Why should you get the same thing when you expand along any row or column? I think you should regard this claim that you always get the same answer by picking any row or column with considerable skepticism. It is incredible and not at all obvious. However, it requires a little effort to establish it. This is done in the section on the theory of the determinant which follows.
Notwithstanding the difficulties involved in using the method of Laplace expansion, certain types of matrices are very easy to deal with.
Definition 3.1.5 A matrix M, is upper triangular if M_{ij} = 0 whenever i > j. Thus such a matrix equals zero below the main diagonal, the entries of the form M_{ii}, as shown.

A lower triangular matrix is defined similarly as a matrix for which all entries above the main diagonal are equal to zero.
You should verify the following using the above theorem on Laplace expansion.
Corollary 3.1.6 Let M be an upper (lower) triangular matrix. Then det
Proof: The corollary is true if the matrix is one to one. Suppose it is n×n. Then the matrix is of the form

where M_{1} is
Example 3.1.7 Let

Find det
From the above corollary, this is −6.
There are many properties satisfied by determinants. Some of the most important are listed in the following theorem.
Theorem 3.1.8 If two rows or two columns in an n×n matrix A are switched, the determinant of the resulting matrix equals

where the i^{th} row of A_{1} is

and if A is an n × n matrix, then

This theorem implies the following corollary which gives a way to find determinants. As I pointed out above, the method of Laplace expansion will not be practical for any matrix of large size.
Corollary 3.1.9 Let A be an n×n matrix and let B be the matrix obtained by replacing the i^{th} row (column) of A with the sum of the i^{th} row (column) added to a multiple of another row (column). Then det
Here is an example which shows how to use this corollary to find a determinant.
Example 3.1.10 Find the determinant of the matrix

First take −1 times the first row and add to the second and the third. The resulting matrix is

It has the same determinant as the original matrix. Next switch the bottom two rows to get

It has determinant which is −1 times the determinant of the original matrix. Hence the original matrix has determinant equal to 1.
The theorem about expanding a matrix along any row or column also provides a way to give a formula for the inverse of a matrix. Recall the definition of the inverse of a matrix in Definition 2.1.22 on Page 130. The following theorem gives a formula for the inverse of a matrix. It is proved in the next section.
Theorem 3.1.11 A^{−1} exists if and only if det(A)≠0. If det(A)≠0, then A^{−1} =

for cof
Theorem 3.1.11 says that to find the inverse, take the transpose of the cofactor matrix and divide by the determinant. The transpose of the cofactor matrix is called the adjugate or sometimes the classical adjoint of the matrix A. It is an abomination to call it the adjoint although you do sometimes see it referred to in this way. In words, A^{−1} is equal to one over the determinant of A times the adjugate matrix of A.
Example 3.1.12 Find the inverse of the matrix

First find the determinant of this matrix. This is seen to be 12. The cofactor matrix of A is

Each entry of A was replaced by its cofactor. Therefore, from the above theorem, the inverse of A should equal

This way of finding inverses is especially useful in the case where it is desired to find the inverse of a matrix whose entries are functions.
Example 3.1.13 Suppose

Find A
First note det

This formula for the inverse also implies a famous procedure known as Cramer’s rule. Cramer’s rule gives a formula for the solutions, x, to a system of equations, Ax = y.
In case you are solving a system of equations, Ax = y for x, it follows that if A^{−1} exists,

thus solving the system. Now in the case that A^{−1} exists, there is a formula for A^{−1} given above. Using this formula,

By the formula for the expansion of a determinant along a column,

where here the i^{th} column of A is replaced with the column vector,
Procedure 3.1.14 Suppose A is an n × n matrix and it is desired to solve the system Ax = y,y =

where A_{i} is obtained from A by replacing the i^{th} column of A with the column
The following theorem is of fundamental importance and ties together many of the ideas presented above. It is proved in the next section.
Theorem 3.1.15 Let A be an n × n matrix. Then the following are equivalent.