# Determinants of Matrices

## Determinants of Matrices - How it Works - Video

### Determinant Formula 2 x 2 and 3 x 3

Determinant Formula 2 x 2:

The determinant of a 2 x 2 matrix is multiplying the top left and the bottom right and subtracting the multiplication of the top right and the bottom left.

The determinant of a 3 x 3 matrix is separating the determinant into 3 smaller determinants. Once that happens, then we can follow the rules of a 2 x 2 matrix. I wouldn't try to remember the formula with all the letters. I would just remember the pattern. After a few goes, it will be easier.

Just be aware that the straight lines do not mean absolute value.

The sign of the determinant tells us the orientation of the column vectors.

What happens when the determinant is positive. A positive determinant means the the column vectors have ant-clockwise orientation.

What happens when the determinant is negative. A negative determinant means that the column vectors have a clockwise orientation.

### Determinant 3 x 3

Determinant 3 x 3:

The first row is the formula for any determinant. You might see a minus sign in front of b1,2 and it means the same. The reason there is a plus sign is that we can change the formula depending how we break down the 3 x 3 matrix.

The determinant with the plus and minus signs tell us the signs that we will use when we multiply the the smaller determinants.

For instance, if we choose the first row to break down the 3 x 3 determinant then the element out in front the first smaller will be positive, the next number out in front will be negative and the last number out in front will be positive.

For instance, if we choose the second row to break down the 3 x 3 determinant then the element out in front the first smaller will be negative, the next number out in front will be positive and the last number out in front will be negative. And so on.

How do we know that? Let's read the next bit.

### Determinant 3 x 3 - Breaking it Down

Determinant 3 x 3 - Breaking it Down:

In the first example we decided to break down the determinant using the first row. We need to break it down to a minor and a cofactor. Our minor is the determinant of the numbers that are remaining. Our cofactor is multiplying 1 by the minor. How do we get 1? Well, we use a1,1 and add the i and j components of our element and i = 1 and j = 1 and the result of 1 + 1 = 2. So (-1)1+1 ==> (-1)2 ==> 1. If we take a look at the plus/minus determinant chart the top left is +.

In the first example we decided to break down the determinant using the second row. We need to break it down to a minor and a cofactor. Our minor is the determinant of the numbers that are remaining. Our cofactor is multiplying -1 by the minor. How do we get -1? Well, we use b1,2 and add the i and j components of our element and i = 1 and j = 2 and the result of 1 + 2 = 3. So (-1)1+2 ==> (-1)3 ==> -1. If we take a look at the plus/minus determinant chart the top middle is -.

In the second example we decided to break down the determinant using the second row. We need to break it down to a minor and a cofactor. Our minor is the determinant of the numbers that are remaining. Our cofactor is multiplying -1 by the minor. How do we get -1? Well, we use c2,3 and add the i and j components of our element and i = 2 and j = 3 and the result of 2 + 3 = 5. So (-1)2+3 ==> (-1)5 ==> -1. If we take a look at the plus/minus determinant chart the middle right is -.

Why is this important? We can find the determinant faster if are elements are a bunch of zeros. Remember that we multiply the elements by the smaller determinants. And anything times 0 is 0. Less work for us, if we choose a row that has a bunch of zeros. Key point here, once you choose a row to break it down, you cannot change it.

### Determinant 4 x 4

Determinant 4 x 4:

Here we have 4 x 4 determinant. We have to break it down to four 3 x 3 determinants and each 3 x 3 determinant we will have to break it down 3 times. Remember to use the plus/minus determinant so that you have the correct sign out in front.

### Example 1

Example 1:

Let's name this matrix A. Our first step is to multiply diagonally and then subtract the multiplication of the other diagonal.

==> 1 * -2 - 5 * 4

==> -2 - 20

==> -22

The determinant of this matrix is -22.

### Example 2 - Part 1

Example 2 - Part 1:

Here we chose the first row to break down the 3 x 3 matrix. Was it the faster choice? We will find out.

So the numbers that we will multiply the 2 x 2 determinants will be 2, 4, and 1. Since we chose the first row, our signs will be +, -, and +.

2 will be multiplied by positive, because 2 is the element a1,1 and (-1)1+1 ==> (-1)2==> 1.

4 will be multiplied by negative, because 4 is the element a1,2 and (-1)1+2 ==> (-1)3==> -1.

1 will be multiplied by positive, because 2 is the element a1,3 and (-1)1+3 ==> (-1)4==> 1.

### Example 2 - Part 2

Example 2 - Part 2:

Here we have broken down each of the 2 x 2 determinants so we have

==> 2 * [0 * 5 - 0 * (-2)] - (-4) * [(-1) * 5 - 0 * 3] + 1 * [(-1) * (-2) - 0 * 3] - Break down the 2 x 2 determinants

==> 2 * [0 - 0] - 4 * [(-5) - 0] + 1 * [2 - 0] - Multiply inside the brackets

==>2 * 0 - 4 * (-5) + 1 * 2 - Subtract inside the brackets

==> 0 + 20 + 2 - Multiply

So 22 is the determinant of our matrix.

### Example 3 - Part 3

Example 3 - Part 1:

Here we chose the second row to break down the 3 x 3 matrix, because it has 2 zeros. Was it the faster choice? We will find out.

So the numbers that we will multiply the 2 x 2 determinants will be -1, 0, and 0. Since we chose the first row, our signs will be +, -, and +.

-1 will be multiplied by negative 1, because 2 is the element a2,1 and (-1)2+1 ==> (-1)3==> -1.

0 will be multiplied by positive 1, because 4 is the element a2,2 and (-1)2+2 ==> (-1)4==> 1.

0 will be multiplied by negative 1, because 2 is the element a2,3 and (-1)2+3 ==> (-1)5==> -1.

### Example 3 - Part 2

Example 3 - Part 2:

Here we have broken down each of the 2 x 2 determinants so we have

==> 1* [4 * 5 - 1 * (-2)] + 0 * [2 * 5 - 1 * 3] + 0 * [2 * (-2) - 4 * 3] - Break down the 2 x 2 determinants

==> 1 * [20 + 2] + 0 * [10 - 3] + 0 * [-4 - 12] - Multiply inside the brackets

==>1 * 22 + 0 * 7 + 0 * (-16) - Subtract inside the brackets

==> 22 + 0 + 0 - Multiply