How to Find the Inverse of a Matrix

Think back to when you first learned about how to solve linear equations. If you were given something like 3x = 6, you would solve by dividing both sides by 3. Since multiplying by is the same as dividing by 3, you could also multiply both sides by to get the same answer: x = 2.

If you needed to solve something like , you could still divide both sides by , but it was probably easier to multiply both side by . The reciprocal fraction is the inverse of because, if you multiply the two fractions, you get 1, which is, in this context, called the (multiplicative) identity; 1 is called the multiplicative identity because multiplying something by 1 doesn't change its value.

This terminology and these facts are very important for matrices. If you are given a matrix equation like AX = C, where you are given A and C and are told to figure out X, you would like to divide off the matrix A. But you can't do division with matrices.

What is an inverse matrix, in simple words?

In simple terms, an inverse matrix is the square matrix A⁻¹ that you can multiply on either side of matrix A to get the identity matrix I. In other words, given matrix A, its inverse matrix A⁻¹ obeys the following:

Why would you want to invert a matrix? What are they for?

Inverse matrices allow you to solve matrix equations in much the same way as inverse fractions allow you to solve one-step (multiplication) linear equations, such as .

Given the matrix equation AX = C, what if you could find the inverse of A, something similar to finding reciprocal fractions for solving linear equations? You can use the inverse of A, written as A⁻¹ and pronounced "A inverse", to cancel off the A from the matrix equation; this then allows you then to solve the matrix equation for X.

How did A⁻¹AX on the left-hand side of the equation (in the second line above) turn into X in the last line? It matches the nature of inverses for regular numbers. If you have a number (such as ) and its inverse (in this case, ) and you multiply them, you get 1. And 1 is the multiplicative identity, so called because 1x = x for any number x.

Inversion works the same way for matrices. If you multiply a matrix (such as A) and its inverse (in this case, A⁻¹), you get the identity matrix I, which is the matrix analog of the number 1. And the point of the identity matrix is that IX = X for any matrix X (meaning "any matrix of the correct size", of course).

It should be noted that the order in the multiplication above is important and is not at all arbitrary. Recall that, for matrices, multiplication is not commutative. That is, AB is almost never equal to BA.

So multiplying the matrix equation "on the left" (to get A⁻¹AX) is not at all the same thing as multiplying "on the right" (to get AXA⁻¹). The product AXA⁻¹ does not equal A⁻¹AX, because you can't switch around the order in matrix multiplication.

Thus, to solve the matrix equation AX=C, you have to multiply A⁻¹ on the left, putting it right next to the A in the original matrix equation. And since you have to do the same thing to both sides of an equation when you're solving, you must multiply "on the left" on the right-hand side of the equation as well, resulting in A⁻¹C.

You cannot be casual with your placement of the matrices; you must be precise, correct, and consistent. This is the only way to successfully cancel off A and solve the matrix equation.

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As you have seen above, inverse matrices can be very useful for solving matrix equations. But—

How do you find the inverse of a matrix?

To find the inverse of a matrix, follow these steps:

1. Write out the matrix that you're wanting to invert.
2. Append to this matrix the identity matrix, making one matrix that is now twice as wide as it is tall.
3. Using row operations, convert the left-hand half of the double-wide in the identity matrix.
4. The new right-hand side of the double-wide is the inverse of the original matrix.

This technique for inverting matrices is kind of clever. Here's an example of how it works:

• Find the inverse of

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How can you know that this matrix is the inverse?

Note that we can confirm that this matrix is the inverse of A by multiplying the two matrices and seeing that we get the identity.

Since the multiplication ended with the identity matrix, the matrix we found is confirmed to be the inverse of the original matrix that they gave us.

Be advised that, in "real life", the inverse is rarely a matrix filled with nice neat whole numbers like this. With any luck, though, especially if you're doing inverses by hand, you'll be given nice ones like this to do.

Is there a formula for inverting a 2x2 matrix?

To find the inverse of a 2-by-2 matrix, use the following formula:

Is there a formula for inverting a 3x3 matrix?

There is a formula, sort of, for the inverse of a 3-by-3 matrix, but it's arguably not the quickest way to proceed. Use the method above instead.

Are there other ways to find the inverse of a matrix?

There are loads of ways to find the inverse of a matrix; Wikipedia gives an extensive list (link). Following the swap-the-identity-matrix method above is probably the easiest of the listed methods. But, in real life (as a mathematician or scientist), other solution methods are preferred. *Any* method is preferred to inverse matrices.