of a matrix
is simply a number that describes that matrix. Only square
matrices can have a determinant.
One of the original uses of a determinant of a matrix
had to do with solving a system of linear equations
. Once the determinant of a matrix
had been calculated, that number could be used to help quickly compute the solution
to a system
of equations. Since that time other applications of determinants have come into play. Some of these include finding the area of a triangle
(and by extension, area
of triangular regions of land), whether points are collinear, and when working with cross products
First we will look at calculating a determinant by hand for a matrix
and for a
matrix. Calculating determinants of higher order square
matrices is more complicated, so we will only focus on these two types of matrices. Calculating the determinant of a matrix
Suppose you have matrix . To find the determinant, you simply take . The notation for the determinant of matrix A can either be or det(A). In the examples below, both notations will be used. Calculating the determinant of a matrix
- If find the determinant of A.
Using the formula and applying it to this matrix gives
- If find the determinant of B.
Again, applying the formula will give
- If find the determinant of C.
In this case we have Notice that we were able to compute determinants that were positive, negative, and zero. It is important to note that while it is possible to obtain a determinant value of zero, there are other implications that arise. A determinant of zero leads to problems in finding an inverse of a matrix and in finding a solution to a system of equations. So while it is mathematically possible to compute a determinant of zero, be aware that is a situation which will cause complications in other computations.
The determinant of a matrix is found by “reducing” it to a series of matrices. Using the calculator to compute determinants
Let’s look at .
We’re going to create three different matrices and find their determinants.
First consider element a in the matrix. If we eliminate the column and row that a appears in, we are left with a matrix that is and its determinant is . We then take that value and multiply it by a. This gives us our first term in computing the determinant of A.
Let’s move on to our second computation. If we consider the element b in the matrix and eliminate its row and column, we are left with a matrix of . The determinant of that matrix is . We then multiply that value by b and get the second term in computing the determinant of A.
- For now, let’s call this value p = a(ei - fh).
Our third, and last, computation comes from considering element c and removing the row and column it appears in to get which has a determinant of . As before, we’ll multiply that value by c to get the last term in our computation of the determinant of A.
- Let’s call this value q = b(di - fg).
However, it’s not just enough to compute p, q, and r. To finally obtain or det(A) we have to add and subtract those in just the right order:
- We’ll call it r = c(dh - eg).
- = det(A) = + (p) - (q) + (r)
You may be thinking that you can never remember all of this. Notice that you weren’t given a formula for this process, but a series of steps to follow. Once you start working through the repetitive process, you won’t be quite so intimidated by all the steps.
- If find the determinant of D.
We’ll start by eliminating the row and column associated with 1. It should be pointed out that this process of using the smaller matrices from a matrix can be done somewhat differently but with the same results. Some books will emphasize an alternative strategy to save steps if it is possible to do so. Rather than learn a variety of methods that may save one step along the way, it is usually best to find one method and stick with it. Because there are so many little steps and so many places to make arithmetic errors, the more automated you can make the task, the more likely you are to find the correct answer without making mistakes.
The matrix that remains will be Now eliminate the row and column associated with the 2.
which has a determinant of .
When we multiply -1 by 1, we get -1.
This leaves and Next, eliminate the row and column associated with the -1.
its determinant of .
Multiply -5 by 2 and you have -10.
This leaves and Finally, take each of the three values you computed to find the answer.
its determinant of .
Multiply 4 by -1 to come up with -4.
= det(D) = + (-1) - (-10) + (- 4) = -1 +10 - 4 = 5
- If find the determinant of E.
We will follow the same process as we did in Example 4.
Start with the 2, eliminate its row and column,
leaving the matrix which has Our next step is to eliminate the row and column associated with the 0 and find the determinant of the resulting matrix. But remember that once we find its determinant,
a determinant of
Then multiply 15 by 2 to get 30.
we’ll multiply it by 0 which will always give 0. So be aware that if you have a zero in the top row of your matrix, you can save yourself one of the determinant combinations because the final result will always be 0.
We now need to eliminate the column and row associated with 1.
This gives us the matrix which has Finally use our three values (remember the second value was 0) and get
a determinant of
Multiply -12 by 1 and we have -12.
= det(E) = + (30) - (0) + (-12) = 30 - 12 = 18
Using the calculator to computer determinants requires that you be able to input a matrix into the calculator. If you help doing this, click here. Let’s re-visit each of the matrices we did in the previous examples and verify that our hand computations were done correctly.
input in the calculator looks like
To obtain the determinant, go to your matrix
menu and then choose the MATH menu. The determinant is the first option on that screen.
When you press
, the determinant command will appear on your home screen and is waiting for you to tell it which matrix
you want to find the determinant of.
Go back to your matrix
screen and choose the name of the matrix
you want to find the determinant of and it will be pasted into your home screen.
When you press
now, the calculator will give you the determinant of matrix