A determinant is based on a square matrix, but the determinant is done up in absolute-value bars instead of square brackets. There is a lot that you can do with (and learn from) determinants, but you'll likely need to wait for an advanced course to learn about them. In this lesson, I'll just show you how to compute 2×2 and 3×3 determinants. (It is possible to compute larger determinants, but the process is much more complicated, so I won't bother with that here.)

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Given a matrix named "*B*", the determinant of *B* is denoted by "det(*B*)", pronounced as "the determinant of *B*", or just "det-bee".

A determinant can be derived only from a square matrix. (Some people have tried to define various pseudo-determinants for non-square matrices, but I don't think they're catching on. All you'll ever hear of will be determinants for square matrices. Because reasons.) If your matrix isn't square, it doesn't have a determinant.

If you have a square matrix, its determinant is written by taking the same grid of numbers, removing the square brackets "[]", and replacing those brackets with absolute-value bars "||", as shown below:

If this is

"the matrix *A*"...

...then this is "the determinant of *A*"

(To type absolute-value bars, use the "pipe" character, which is probably on the same keyboard key as the "backslash" character.)

Just as absolute values can be evaluated and simplified to get a single number, so can determinants. The process for evaluating determinants is pretty messy, so let's start simple, with the 2×2 case.

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For a 2×2 matrix, its determinant is found by subtracting the products of its diagonals, which is a fancy way of saying in words what the following says in pictures:

the matrix *A* with variables:

the determinant of *A* (or "det *A*"):

the matrix *A* with numbers:

the determinant of *A* (or "det *A*"):

In other words, to take the determinant of a 2×2 matrix, you follow these steps:

- Multiply the values along the top-left to bottom-right diagonal
- Multiply the values along the bottom-left to top-right diagonal
- Subtract the second product from the first
- Simplify to get the value of the 2-by-2 determinant

"But wait!" I hear you cry; "Aren't absolute values always supposed to be positive? The numerical matrix above is shown as having a negative determinant. What's up with that?" You make a good point.

Yes, determinants can be negative! Determinants are similar to absolute values, and use the same notation, but they are not identical, and one of the differences is that determinants can indeed be negative.

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- Evaluate the following determinant:

In this exercise, they've given me a determinant (rather than a matrix), so I can get right to work. I multiply the diagonals (highlighted with purple arrows in my working below), and subtract:

Then my answer is:

det(*A*) = 3

- Find the determinant of the following matrix:

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Here, they've given me a matrix, and asked me to find the determinant of it.

First, I'll convert from a matrix to a determinant by swapping out the brackets for absolute-value bars. Then I'll multiply along the diagonals (blue arrows below), subtract the products, and simplify to get my numerical answer:

Then my answer is:

det(*A*) = 7

That's really all there is to 2-by-2 determinants. Just make sure you multiply and subtract in the right order, and you'll be fine.

You can use the Mathway widget below to practice finding the values of the determinants of 2-by-2 matrices (or skip the widget and continue on to the next page). Try the entered exercise, or type in your own exercise. (Use their "matrix" button to enter your own matrix, or else use the bracket notation "[[a b],[c d]]. If you omit the comma, the software won't understand what you mean.) Then click the button and select "Find the determinant" to compare your answer to Mathway's.

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*(Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.)*

URL: https://www.purplemath.com/modules/determs.htm

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