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How to Take a Determinant of a Matrix

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  • 0:06 What Was a Matrix?
  • 1:15 Square Matrices
  • 2:39 The Determinant of a…
  • 4:15 Calculating Larger Matrices
  • 6:03 Summary
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Lesson Transcript
Instructor: Robert Egan
Matrices are incredibly powerful and can help you do all sorts of things, but one of the most basic (and surprisingly helpful) operations you can perform on one is to take its determinant. Learn how to do that here!

What's a Matrix?

Example of an MxN matrix
MxN Matrix Example

One of the simplest, most unique operations you can perform with a matrix is called the determinant. In this video, we're going to concentrate on what it looks like and how you calculate it. We're not going to go too in depth but a little matrix basics and vocabulary is needed for us to this.

Matrices are grids of numbers that are surrounded by brackets, a kind of squared-off parentheses. And matrices are labeled by their number of rows and columns, so we call a matrix an M x N matrix, where M is the number of rows and N is the number of columns. What you see here is a 3 by 5 (3 x 5) matrix, because it has 3 rows and 5 columns of numbers which is a total of 15 numbers inside. This would be a 6 by 1 (6 x 1) matrix because of its 6 rows and only 1 column. Both of these examples look like rectangles because the number of rows and columns are different.

Square Matrices

When the number of rows and columns are the same, we end up with a square matrix. That means that we could have a square matrix that is 1x1 or 2x2 or 3X3. Because they have the same number of rows and columns, the mathematical way of defining a square matrix is to say that it is any matrix that is N x N.

A square matrix is known as an NxN matrix in mathematics
Square Matrix

The reason we need to know what a square matrix is, is that a determinant can only be calculated on square matrices. As long as it is a square matrix, the determinant of any matrix is simply a number, so the determinant takes a matrix and boils it down to one value. Actually calculating the determinant gets more and more difficult as the matrix gets bigger and bigger, so we'll start by looking at the smallest square matrix (1x1) and practice on bigger and bigger ones.

Taking the determinant of a 1x1 matrix is what mathematicians call 'trivial.' This is because there is essentially nothing to do. For example, the determinant of the matrix (which is denoted by the straight bars instead of the brackets) |4| is just 4. Or the determinant of the matrix |-5| is just -5. Notice that the matrix is no longer in brackets but just in straight, up-and-down bars, kind of like absolute value signs. Although the | | symbols look the same as an absolute value, this is just a coincidence and probably comes down to the fact that mathematicians are lazy and don't want to draw anything else more complicated.

2X2 Matrix

Taking the determinant of a 2x2 matrix starts to get a little more complicated. Let's look at the example |5 4 , 8 2|. Again, because the matrix is surrounded by | | we know that we are being asked to calculate the numeric value of the determinant. Now, we have to start with a process that involves multiplying the diagonals and subtracting the products. We start with the top left to bottom right diagonal; in this example, we would do (5)(2). We subtract the bottom left to top right diagonal, which is (8)(4) from that. What we end up with is 10 - 32 = -22, so the determinant of this matrix is simply -22.

Multiply the diagnonals and substract the products to find the determinant
Determinant in 2x2 Matrix

Another quick example might be |-1 5 , -4 3|. Here we would do (-1)(3) - (-4)(5). Be careful when you subtract a negative, since it changes sign. So, (-1)(3)-(-4)(5) = -3 + 20 = 17.

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