Let A be an n×n matrix. The determinant of A, denoted as det
Definition 21.1.1 Let A =

Example 21.1.2 Find det
From the definition this is just
Having defined what is meant by the determinant of a 2 × 2 matrix, what about a 3 × 3 matrix?
Definition 21.1.3 Suppose A is a 3 × 3 matrix. The ij^{th} minor, denoted as minor
Example 21.1.4 Consider the matrix

The

The

Definition 21.1.5 Suppose A is a 3 × 3 matrix. The ij^{th} cofactor is defined to be
Example 21.1.6 Consider the matrix

The

It follows

The

Therefore,

Similarly,

Definition 21.1.7 The determinant of a 3×3 matrix A, is obtained by picking a row (column) and taking the product of each entry in that row (column) with its cofactor and adding these up. This process when applied to the i^{th} row (column) is known as expanding the determinant along the i^{th} row (column).
Example 21.1.8 Find the determinant of

Here is how it is done by “expanding along the first column”.

You see, we just followed the rule in the above definition. We took the 1 in the first column and multiplied it by its cofactor, the 4 in the first column and multiplied it by its cofactor, and the 3 in the first column and multiplied it by its cofactor. Then we added these numbers together.
You could also expand the determinant along the second row as follows.

Observe this gives the same number. You should try expanding along other rows and columns. If you don’t make any mistakes, you will always get the same answer.
What about a 4 × 4 matrix? You know now how to find the determinant of a 3 × 3 matrix. The pattern is the same.
Definition 21.1.9 Suppose A is a 4 × 4 matrix. The ij^{th} minor is the determinant of the 3 × 3 matrix you obtain when you delete the i^{th} row and the j^{th} column. The ij^{th} cofactor, cof
Definition 21.1.10 The determinant of a 4 × 4 matrix A, is obtained by picking a row (column) and taking the product of each entry in that row (column) with its cofactor and adding these together. This process when applied to the i^{th} row (column) is known as expanding the determinant along the i^{th} row (column).
Example 21.1.11 Find det

As in the case of a 3 × 3 matrix, you can expand this along any row or column. Lets pick the third column. det

Now you know how to expand each of these 3 × 3 matrices along a row or a column. If you do so, you will get −12 assuming you make no mistakes. You could expand this matrix along any row or any column and assuming you make no mistakes, you will always get the same thing which is defined to be the determinant of the matrix A. This method of evaluating a determinant by expanding along a row or a column is called the method of Laplace expansion.
Note that each of the four terms above involves three terms consisting of determinants of 2 × 2 matrices and each of these will need 2 terms. Therefore, there will be 4 × 3 × 2 = 24 terms to evaluate in order to find the determinant using the method of Laplace expansion. Suppose now you have a 10 × 10 matrix and you follow the above pattern for evaluating determinants. By analogy to the above, there will be 10! = 3,628,800 terms involved in the evaluation of such a determinant by Laplace expansion along a row or column. This is a lot of terms.
In addition to the difficulties just discussed, you should regard the above 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.
With this definition of the cofactor matrix, here is how to define the determinant of an n × n matrix.
Definition 21.1.13 Let A be an n × n matrix where n ≥ 2 and suppose the determinant of an
 (21.1) 
The first formula consists of expanding the determinant along the i^{th} row and the second expands the determinant along the j^{th} column.
Theorem 21.1.14 Expanding the n × n matrix along any row or column always gives the same answer so the above definition is a good definition.