How to Find the Determinant of a 4x4 Matrix

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  • 0:03 Setting Up the Problem
  • 4:32 Solution to the Problem
  • 4:48 Checking Your Work
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Lesson Transcript
Instructor: Sharon Linde

Sharon has a Masters of Science in Mathematics

Finding the determinant of a matrix helps you do many other useful things with that matrix. This lesson shows step by step how to find a determinant for a 4x4 matrix. The process used is applicable to a square matrix of any size larger than 2x2.

Setting Up the Problem

Henry is a graduate student who's been working for months on a problem in finance theory. He views the results of his latest data in an email on his phone while he's backpacking. He's left with a set of four equations and four unknowns that he can set up in a 4 x 4 matrix as shown here:

determinant of 4x4 matrix

Henry is very, very interested to know what the determinant of this matrix is. What is a determinant, and why is Henry so interested? A determinant is a single, specific number associated with a square matrix. If the determinant of this matrix is equal to zero, it will mean at least another few months of research. But if the determinant is anything other than zero, he will have made a significant discovery in his field and will be able to earn his degree within the next year.

Henry tries several times to navigate to a determinant calculator site, which would answer his question in a few seconds, but the signal is too weak and keeps crashing. Eventually his battery dies, leaving him only with the option of calculating the determinant by hand. He groans and gets out his pencil and paper. Let's take a closer look at how he finds the determinant.

There are a number of different ways to find the determinant of a 4 x 4 matrix, but we'll show you how to do it by using expansion along any row or column of a matrix. When you've chosen which column or row to expand, then the determinant is simply each element in that row or column multiplied by the determinant of that element's sub-matrix with the appropriate sign. A sub-matrix is the portion of the original matrix that does not include the elements in the same row or column as the current element. This sounds a bit confusing, but is actually pretty easy once you see how it's done. We'll get to the first example shortly.

First, we need to talk about the appropriate sign we just mentioned. In the matrix here, we've shown the signs associated with the elements being expanded.

determinant of 4x4 matrix

For example, if you were to expand the top row of a 4 x 4 matrix, the first and third elements in the row would be multiplied by a +1, while the second and fourth elements would be multiplied by -1.

Now that we understand those signs and what a sub-matrix is, we can choose a row or column to expand. But which one would be easiest to expand? It turns out that the row or column with the most zeroes in it makes for the fewest computations using the expansion method. Looking at our matrix, the third column has two zeros out of four elements. This means our work will be cut in half if we chose that method.

Now that we've chosen a column, we can expand along that column. How do we do that? First, let's show what we are going to do using words, and then we'll see the sub-matrix determinants written out.

To expand along the third column, we will have four terms:

(+1)(3) multiplied by (determinant of submatrix of the element in the first row, third column)

(-1)(0) multiplied by (determinant of submatrix of the element in the second row, third column)

(+1)(-2) multiplied by (determinant of submatrix of the element in the third row, third column)

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