Matrix Determinants

This is my first video lesson, and it will help you calculate the determinant of a matrix. I learned to do this in the FE refresher course I am taking and I am confident that if you can master this topic, it'll be easy points on the Fundamentals of Engineering exam. Check it out, and let me know what you think!

The general procedure for this is to draw diagonal lines across the rows, multiply the numbers along each line, and add or subtract these products. Products on lines going from the top-left to bottom-right are added; from top-right to bottom-left are subtracted. If you're familiar with vector operations, this is similar to the cross product.


For a 2x2 matrix, we simply draw an "x". Multiply the top left and the bottom right. From that, subtract the product of the top right and the bottom left. Viola. You've got it.

A 3x3 is a little more tricky. To the right of the matrix, copy down the first three columns again -- so you have five columns total. Then start drawing the diagonal lines downward starting from the top-left corner. You should have three full diagonals. Then do the same starting at the bottom-left and going diagonally upward. You'll have three of these for six total. (This is hard to explain verbally, which is why I made the video!) Then find all the products and sum them up -- the products on the lines going from top to bottom are positive and the ones going from bottom to top are negative.

Watch out! I'm told that on the FE exam, often one of the incorrect choices will be an incorrect answer that you would get from a commonly-made mistake. In this case, be sure to subtract the proper diagonals and keep your signs straight.

Once you get into the 4x4 and beyond, it gets much more complicated. I am not anticipating that this question will be on your exam since this is a timed test; if it were, I would either put it into my calculator, or skip it and move on to another one as it will not likely be worth the time to solve.

The determinant of a non-square matrix -- say, 2x3 -- is undefined. The determinant of a 1x1 matrix -- say | 3 | -- is just the single digit present (in this case, 3).

Below are some practice problems, similar to what I think you might find on the FE exam. If you need more problems, just make some up -- you now know how to solve for the determinant of any 2x2 or 3x3 matrix, so you can invent your own matrices and check your solutions with this matrix determinant calculator I found on the web. Practice practice practice!

Practice problems:

Problem a)
Find the determinant of the following matrix:

|3 4|
|5 6|

Problem b)
Find the determinant of the following matrix:

| 1 4 6 |
|-1 0 2 |
| 0 5 9 |

Solutions:
Highlight the text below to reveal the answers.

Problem a) -2
Problem b) -4