# Determinant of a 2×2 Matrix

Suppose we are given a square matrix $A$ with four elements: $a$, $b$, $c$, and $d$.

The determinant of matrix A is calculated as

If you can’t see the pattern yet, this is how it looks when the elements of the matrix are color coded.

• We take the product of the elements from top left to bottom right, then subtract by the product of the elements from top right to bottom left.

## The Determinant of 2 x 2 Matrix (animated)

### Examples of How to Find the Determinant of a 2×2 Matrix

Example 1: Find the determinant of the matrix below.

This is an example where all elements of the 2×2 matrix are positive.

Example 2: Find the determinant of the matrix below.

Here is an example when all elements are negative. Make sure to apply the basic rules when multiplying integers. Remember, the product of numbers with the same sign will always be positive. In contrary, if the signs are different the product will be negative.

Example 3: Evaluate the determinant of the matrix below.

Make sure to remember the rules on how to subtract integers. That is, when you subtract integers, you change the operation from subtraction to addition but you must switch the sign of the number directly found to its right (it’s called the subtrahend) then proceed with regular integer addition.

Example 4: Evaluate the determinant of the matrix below.

You may also encounter a problem where some of the elements in the matrix are variables. Treat this just like a normal determinant problem. Plug those variables in the designated spots in the formula then simplify as usual.

Example 5: Find the value of $x$ in the matrix below if its determinant has a value of $-12$.

This is not a “trick” question. We can actually find the value of $x$ such that when we apply the formula we get $-12$.

Get the determinant of the given matrix then set it equal to $-12$. By doing so, we generate a simple linear equation that is solvable for $x$.

Replace $x$ by $7$, then calculate the determinant. We expect to get $-12$.