To find the determinant of a $3 \times 3$ matrix, we can use a simple formula.

Imagine we have a matrix like this:

$$A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}.$$

To find the determinant, we follow this formula:

$$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg).$$

Here’s How to Do It Step by Step:

1. Multiply the numbers in the first row by special numbers called cofactors.

2. Add up the results. Remember to pay attention to the plus and minus signs!

Let’s Try an Example:

For our matrix,

$$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix},$$

we'll find the determinant like this:

1. First, do the math inside the parentheses:

   • For $a = 1$:

     • Calculate $5 \cdot 9 - 6 \cdot 8$
     • This gives us $45 - 48 = -3$.

   • For $b = 2$:

     • Calculate $4 \cdot 9 - 6 \cdot 7$
     • That gives us $36 - 42 = -6$.

   • For $c = 3$:

     • Calculate $4 \cdot 8 - 5 \cdot 7$
     • This results in $32 - 35 = -3$.

2. Now plug these back into the main formula:

   • $1 \cdot (-3) - 2 \cdot (-6) + 3 \cdot (-3)$

3. Work this out step by step:

   • $1 \cdot (-3) = -3$.
   • $-2 \cdot (-6) = +12$.
   • $3 \cdot (-3) = -9$.

4. Now add those values together:

   $-3 + 12 - 9 = 0.$

So, the determinant of matrix $A$ is

$\text{det}(A) = 0!$

That's it! You have calculated a determinant of a $3 \times 3$ matrix!