Lesson Objectives
• Learn how to find the determinant of a 2 x 2 matrix
• Learn how to find the determinant of a 3 x 3 matrix

## How to Find the Determinant for a Matrix

In this lesson, we will focus on how to find the determinant of a 2 x 2 and a 3 x 3 matrix, using the shortcut methods.

### Finding the Determinant of a 2 x 2 matrix

We will notate the determinant of matrix A with vertical bars |A| or det(A). $$A=\left[ \begin{array}{cc}a&b\\ c&d\end{array}\right]$$ $$|A|=a \cdot d - c \cdot b$$ Let's see an example.
Example #1: Find the determinant. $$A=\left[ \begin{array}{cc}4&3\\ -6&2\end{array}\right]$$ $$|A|=4 \cdot 2 - (-6 \cdot 3)$$ $$|A|=8 + 18=26$$

### Finding the Determinant of a 3 x 3 matrix

Finding the determinant for a 3 x 3 matrix is a bit more complex, again we will use the shortcut method. Let's work through an example.
Example #2: Find the determinant. $$A=\left[ \begin{array}{ccc}-4&-5&3\\ 0&1&2\\3&5&0\end{array}\right]$$ Step 1) Copy the first two columns and place them at the end of the matrix. $$\begin{array}{ccc}-4&-5&3&-4&-5\\ 0&1&2&0&1&\\3&5&0&3&5&\end{array}$$ Step 2) Starting at the top left multiply down on each diagonal. This will give you three products. We will find the sum of these three products. $$-4 \cdot 1 \cdot 0=0$$ $$-5 \cdot 2 \cdot 3=-30$$ $$3 \cdot 0 \cdot 5=0$$ $$0 + (-30) + 0=-30$$ Step 3) Repeat this process, but now start at the bottom left and multiply up each diagonal. Again, we want to find the sum of these three products. $$3 \cdot 1 \cdot 3=9$$ $$5 \cdot 2 \cdot -4=-40$$ $$0 \cdot 0 \cdot -5=0$$ $$9 + (-40) + 0=-31$$ Step 4) Subtract away the number found in step 3 from the number found in step 2. $$-30 - (-31)=1$$ $$|A|=1$$

#### Skills Check:

Example #1

Find the determinant. $$A=\left[ \begin{array}{cc}4&-1\\ 2&-3\end{array}\right]$$

A
$$|A|=19$$
B
$$|A|=18$$
C
$$|A|=15$$
D
$$|A|=5$$
E
$$|A|=-10$$

Example #2

Find the determinant. $$A=\left[ \begin{array}{cc}-4&4\\ 1&-2\end{array}\right]$$

A
$$|A|=11$$
B
$$|A|=7$$
C
$$|A|=4$$
D
$$|A|=6$$
E
$$|A|=-2$$

Example #3

Find the determinant. $$A=\left[ \begin{array}{ccc}-3&-3&-4\\ -2&3&3 \\3&1&-1\end{array}\right]$$

A
$$|A|=-41$$
B
$$|A|=41$$
C
$$|A|=13$$
D
$$|A|=12$$
E
$$|A|=55$$