Given a 2x2 matrix
\( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)
The determinant of matrix A is written
\( det(A) = \abs{A} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} \)
Notice we use single vertical bars around the name of the matrix( \( \abs{A}\) ) and we replace the brackets with vertical bars on the matrix itself. This is an important distinction.
\( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)
is a matrix, while
\( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \)
is a determinant.

To calculate the determinant, we multiply the top left times bottom right diagonal and we subtract the other diagonal, top right times bottom left. It looks like this.

It is best not to memorize the formula but to visualize the concept of multiplying the diagonal elements.

In order to calculate the determinant of a 3x3 matrix, we build on the same idea as the determinant of a 2x2 matrix. Before we go through the details, watch this video which contains an excellent explanation of what we discuss here. You will need to work through this concept in your head several times before it becomes clear. So watch this video first and then go through the explanation that follows.

If we have a 3x3 matrix that looks like this
\( \begin{bmatrix} i & j & k \\ a & b & c \\ d & e & f \end{bmatrix} \)
we calculate the determinant by breaking the matrix into 2x2 matrices and calculating the determinant of each of those. It works like this.
Start with the element in the first row and first column, in this case \(i\). We remove the row and column that \(i\) is in and multiply \(i\) by the determinant of the remaining 2x2 matrix. So the first element is
\( i \cdot \begin{vmatrix} b & c \\ e & f \end{vmatrix} \)
Of course, we know how to calculate the determinant of the 2x2 matrix.

We continue this procedure by going across the top row (you can also go down the first column and your answer will be the same but we need to go across the top row for calculating cross products). The second element is
\( j \cdot \begin{vmatrix} a & c \\ d & f \end{vmatrix} \)
and the third element is
\( k \cdot \begin{vmatrix} a & b \\ d & e \end{vmatrix} \)

Notice for each element, we remove the row and column for that item in the first row to get the corresponding 2x2 matrix.

Okay, now that we have all the elements, we need to put them together to get the result. We will show the result and discuss more afterwards.
\( \begin{vmatrix} i & j & k \\ a & b & c \\ d & e & f \end{vmatrix} = i \cdot \begin{vmatrix} b & c \\ e & f \end{vmatrix} - j \cdot \begin{vmatrix} a & c \\ d & f \end{vmatrix} + k \cdot \begin{vmatrix} a & b \\ d & e \end{vmatrix} \)
Notice that the middle element is subtracted but the first and last are added. This is important to obtaining the correct result.

Again, it is important to visualize the concept instead of memorizing equations as you learn how to do this.

Okay, here is a video with an example.

Notes
1. Calculating determinants can be done only on square matrices, i.e. matrices with the same number of rows and columns.
2. If the determinant of a matrix is zero, we say that the matrix is singular.
3. You do not have to go across the top row to form the cofactors like we did in the discussion above. However, there are rules about the sign that goes in front of each term depending on where you start. So, initially, while you are first learning this technique, we recommend that you stick with the first row or column and alternate signs, starting with a positive sign. (Of course, check with your instructor to see what they expect.)
4. For larger matrices (more than 3 rows and columns), we just repeat the procedure, remembering to alternate signs.

An alternate method for calculating the determinant involve calculating across diagonals. This method is kind of a 'shortcut'. So check with your instructor to see what they expect. Even if they don't allow you to use this technique on homework and exams, you can use it to check your work.
Note - You still need to be able to use the cofactor method since cofactors are important in other areas of linear algebra. However, in calculus, either method will suffice.

Here are a few properties of determinants that you can use to help simplify your calculations.

1. Column Factor - - If you have a column that contains a common factor, you can pull that common factor out before calculating the determinant. For example,
\( \begin{vmatrix} 2a & b \\ 2c & d \end{vmatrix} = 2 \begin{vmatrix} a & b \\ c & d \end{vmatrix} \)
Notice that we are saying that the determinants are equal, not the matrices themselves.