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More Determinants: 3x3 Matrices

More Determinants: 3x3 Matrices

Author: c o

To define and demonstrate the application of determinants of 3x3 matrices.

This packet builds on the learner's knowledge of determinants of 2x2 matrices and applies it to the 3x3 case, introducing new concepts where necessary.

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The Determinant And Its Uses

Before you begin, you might want to checkout the lesson on determinants of 2x2 matricies


Finding Inverse Matrices

The properties of determinants that we learned for 2x2 matrices still hold for the 3x3 case, and in fact, they are true for a square matrix of any size.  Hence, the formula for the inverse of a matrix M is still given by M-1 = 1/det(M) * adj(M), where adj(M) is the adjugate matrix. 

Unfortunately, the general definition of the adjugate matrix is somewhat complicated. Formally, the adjugate of a matrix M is the transpose of its cofactor matrix. But what are the transpose and cofactor matrix? 

For starters, the transpose of a matrix M is given by swapping the columns of M with the rows of M, and is denoted Mt.  

For example


and   is   

The cofactor matrix is a little bit more complicated.  The cofactor of a matrix M is a matrix of determinants of smaller matrices.  More specifically

If M is an nXn matrix, then its cofactor matrix has entries (-1)i+jmij, where each mij is the determinant of an (n-1)X(n-1) matrix that is the result of omitting the ith row and the jth column from M.  

All of the above is more complicated to state than it is to calculate, so lets look at the definition of the adjugate matrix for 3x3 matrices below:

Example - Determinant And Adjugate

Since both calculations can seem a little daunting on pen-and-paper, we work through an example of each for a single 3x3 matrix.

We didn't quite finish with finding the adjugate - the next section wraps up.

Errata and a Quick Follow-Up

Errata - Two Things

First, the final calculation of the (3,3) entry in the matrix should have been -3, not -1.  

Second, I forgot to finish finding the adjugate!  We only got as far as the cofactor matrix, the remaining step is to take the transpose of the cofactor matrix.

So, the cofactor matrix was:


taking the transpose of this gives the adjugate matrix



Now that we have found both the determinant of M and the adjugate, we can easily calculate the inverse.  The determinant is -7, so using the formula for the inverse, we have

M-1 = 1/det(M) * adj(M) = 


Cramer's Rule Example

In this video we solve a matrix equation using Cramer's rule.