Cramer's Rule

In practice, Cramer's Rule is a handy way to solve for just one of the variables in the system of equation without having to solve the entire system of equations; in other words, if you only need one variable from a system of, say, five equations in five unknowns, Cramer's Rule allows you to solve for just that one variable.

For some reason, they don't usually teach Cramer's Rule this way; they'll give you a system of equations and have you use Cramer's Rule to solve for each and every variable. But the point of the Rule is that, instead of solving the entire system of equations to make sure you get the value you need, you can instead save time and solve for just the one value that you need.

(I can be a bit dense some times. It wasn't until my physics professor pointed out that "this would be a good time to use that Cramer's Rule that they taught you" that it even dawned on me how helpful the Rule can be.)

How does Cramer's Rule work?

Cramer's Rule tells us to form certain determinants and divide them in order to find variables' values. The denominator of all of the divisions will be the determinant of the coefficient matrix. For the numerators, you will pick a variable; you will replace the coefficients of that variable's entries in the coefficient matrix with the "column vector" containing the constants on the other side of the "equals" sign in the original system of equations.

(I will refer to this column vector as the "answers column" and to the matrix and determinant created by replacing a column with the answers column as the "answers" matrix and determinant. This is not proper terminology, but I can't find any name for these.)

The value for a chosen variable is found by dividing the value of that variable's answers determinant by the value of the coefficient determinant.

To see how Cramer's Rule works, let's apply it to the following system of equations:

We have the left-hand side of the system with the variables; the square matrix containing the variables' coefficients is the coefficient matrix). On the right-hand side, after the "equals" sign, we have the answer values; the width-1 matrix containing these values is what I'm calling the "answer" column..

Let's name the determinant of the coefficient matrix of the above system "D":

The column of answer values on the right-hand side of the "equals" signs in the system of equations above can be made into its own little matrix:

(Technically this is a "column vector", as mentioned above, but you almost certainly won't need to know that terminology right now, if ever. However, if your instructor uses this term, the tall skinny one-column matrix above is what is meant.)

We'll take the coefficient determinant D and replace the first column of values (being the blue coefficients from the x-column of the original system) with the answer values. We'll call the resulting answers determinant "D_x" (pronounced as "dee, sub-eks"), to remind us that it's the x coefficients that we replaced (and also that it's the value of x that we're looking to find):

As you can see, the red values from the answer column have replaced the original blue values from the x-variable terms in the original system of equations.

Similarly, determinants with the other two variables' values replaced are formed and named similarly; namely, the determinants D_y and D_z for this system are:

...where the red answer values have replaced the y-values in the middle green column, and:

...where the red answer values have replaced the z-values in the right-hand purple column.

We can evaluate each of these determinants (using the method explained here). The results are these:

So, now that we have all these determinants and their values, what do we do with them?

How does Cramer's Rule find a variable's value?

Cramer's Rule says that we can find the value of a given variable by dividing that variable's determinant by the regular coefficient-determinant's value. That is, Cramer's Rule specifies this relationship:

Taking the determinant values that we derived from the system they'd given us and doing the relevant divisions, we get these values for the variables:

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And that's all there is to Cramer's Rule. To find whichever variable you want (call it, say, β or ["beta"]), just evaluate the determinant quotient D_β ÷ D.

(Please don't ask me to explain why this works. If you really want to know, there are loads of proofs online, like this and this. Otherwise, just trust me that determinants can work many kinds of magic.)

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What is an example of solving by using Cramer's Rule?

• Given the following system of equations, find the value of z.

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The point of Cramer's Rule is that you don't have to solve the whole system to get the one value you need. This saved me a fair amount of time on some physics tests. I forget what we were working on (something with wires and currents, I think), but Cramer's Rule was so much faster than any other solution method for finding the one value I needed — and God knows I needed the extra time.

Don't let all the subscripts and stuff confuse you; the Rule is really pretty simple. You just pick the variable you want to solve for, replace that variable's column of values in the coefficient determinant with the answer-column's values, evaluate that determinant, and divide by the coefficient determinant. That's all there is to it.

Almost.

What if the coefficient determinant is zero?

You can't divide by zero, so what does this mean if the determinant by which you need to divide evaluates to zero? I won't go into the technicalities here, but "D = 0" means that the system of equations has no unique solution; instead, the system may be inconsistent (that is, it has no solution at all) or dependent (that is, it has an infinite number ofsolutions, which may be expressed as a parametric solution such as "(a, a + 3, a − 4)").

In terms of Cramer's Rule, "D = 0" means that you'll have to use some other method (such as matrix row operations) to solve the system. If D = 0, you can not use Cramer's Rule.