2. Determinant of a matrix

2.1Introduction¶

In the previous chapter we learned that matrices are very similar to numbers - you know how to add two matrices, how to subtract them and you know how to multiply two matrices. The only operation we haven’t mentioned so far is divison - if you have two matrices $A$ and $B$, what would it mean to divide matrix $A$ by the matrix $B$?

In order to answer that quesion, let’s see what does it mean to divide two numbers: for example, if we are given numbers 5 and 7, then to divide number 5 by 7 we have to multiply 5 by $\frac{1}{7},$ i.e.

What is the connection between numbers 7 and $\frac{1}{7}?$ Well, if we multiply them in any order, we get number 1 as a result:

This also means that the reciprocal value or the inverse of the number 7 is $\frac{1}{7}.$

So, we can see that there are two parts that are crucial when dividing number $a$ by the number $b$:

1. Find the inverse $\frac{1}{b}$ of the number $b$

2. Multiply $a$ and $\frac{1}{b}$

We can use this approach to define “divison” of matrices. If we would like to divide matrix $A$ by the matrix $B$, firstly we need to define what would be the inverse of the matrix $B$ and then we would simply need to multiply $A$ by that inverse.

So, if we would like to divide the matrix $A$ by the matrix $B$, we would need to find the inverse $B^{-1}$ of the matrix $B$ and then compute $A \cdot B^{-1}.$ However, here we stumble upon a problem - not all matrices are invertible.

2.2Determinant of a matrix¶

Because not every matrix is invertible, we need a criterion that will allow us to check whether or not a given matrix has an inverse. For that, we will be using the determinant. The determinant is a number that is associated to every square matrix. The determinant of a matrix $A$ is denoted by $\lvert A \rvert$ or by $\text{det}(A).$ The determinant of a $2 \times 2$ matrix is given by a simple formula

In order to find determinant of matrices of higher order, we will use two methods:

• Laplace expansion

• Elementary transformations

Laplace expansion¶

Elementary transformations¶

As we have seen, Laplace expansion is a very useful method of finding the determinant of a matrix that has a lot of zeros along some row or a column, however what if our has matrix doesn’t have any zeros? In that case, we will use elementary transformations to get the zeros in our matrix so that we can use the Laplace expansion to find the determinant. To use an elementary transformation simply means to multiply a row of a matrix by some number and adding it to some other row. Obviously, the end-goal of performing elementary transformations is to get zeros along some column of the given matrix so that we can use Laplace expansion to calculate the determinant.

The most important theorem when talking about the determinant is the Binet-Cauchy theorem.

2.3Matrix inversion¶

As we have mentioned in the introduction, the main point of the determinant is to give us a criterion for checking whether or not a matrix has an inverse or not. That criterion is the following:

Since we have just learned how to compute the determinant of a matrix, we know how to chekc whether or not a matrix has an inverse or not. Next thing we would like to know is how to explicitly compute the inverse of a matrix.

Finally, the inverse of the matrix $A$ is given by the formula

2.4Cramer’s system¶

One of the main application of matrices we are going to see is revolving around solving system’s of linear equations.

In general, every system can be written in matrix form as

where $A$ is the matrix of coefficients, $X$ is the column-matrix of unknowns and $B$ is the column-matrix of the right-hand side of the system.

Cramer’s systems are particularly easy to solve because their solutions are given by the formula

where $D = \text{det}(A)$ and $D_i$ is the determinant of the matrix obtained by replacing the $i$-th column of the matrix $A$ by the column-matrix $B$.