Math Lesson 156: Determinants of Matrices

Finding the determinant of the of any square matrix is simple. All one has to do is rewrite the matrix, except instead of brackets, you have to have straight lines. That’s basically it!

Where the hard part comes in as actually solving the determinants. Mostly, you’ll get either 2*2 determinants or 3*3 determinants. I cover both of these in this post.

If you have a 2*2 matrix that looks like this:

[A_1,1 A_1,2]
[A_2,1 A_2,2]

Then all you have to do is multiply the element in the top left corner, A_1,1, by the bottom right corner, A_2,2 , and from that product subtract the products of the bottom left corner, A_2,1 by the top right corner, A_1,2. The exact formula will look like this:

A_{1,1 *} A _1,2 – A_{2,1 *} A_2,2

But the 3*3 matrix is harder. The first thing that we do is on the right side of the determinant, what we do is rewrite the first two columns. Having done this, we find the sum of the products of multiply by the diagonals, starting from element A_1,1 and going all the way to A_1,3. When we multiply by the diagonals, we must have three multiplicands.

Then, once we’ve done that for the top row, we do the same thing starting from element A_3,1  and multiplying upwards by diagonals, until we get to element A_3,3. This time, the products that you get for the elements on the bottom row are subtracted, not added, in the equation.

In other words, if we have matrix:

[A_1,1 A_1,2 A_1,3]
[A_2,1 A_2,2 A_2,3]
[A_3,1 A_3,2 A_3,3]

Then the way you would solve it would be:

(A_{1,1 *} A_{2,2 *} A_3,3) + (A_{1,2 *} A_{1,2 *} A_2,3) + (A_{1,3 *} A_{2,1 *} A_3,2) – (A_{3,1 *} A_{2,2 *} A_1,3) – (A_{3,2 *} A_{2,3 *} A_1,1) – (A_{3,3 *} A_{2,1 *} A_1,2)

There is another method for solving determinants, although this method is best explained with examples, and not words.

If we have the same matrix:

[A_1,1 A_1,2 A_1,3]
[A_2,1 A_2,2 A_2,3]
[A_3,1 A_3,2 A_3,3]

Then the result will be as follows:
|_{A2,2 A2,3}|           |_{A2,2 A2,3}|          |_{A2,1 A2,2}|
A_1,1*|_{A3,2 A2,3}|- A_{1,2 *} |_{A3,2 A3,3}|+A_1,3*|_{A3,1 A3,2}|

So, the determinants would be solved first, then they would be multiplied by the elements that are in front of them, and then you’d subtract the second term from the first term, and then add the third term.

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Posted in Internet Math, Math, Matrices, Pre-Calculus