Inverse Matrix: Definition, Properties & Formula

Instructor: Sharon Linde
Matrices are powerful tools for a wide variety of applications: computer gaming, massive data visualization, and designing buildings for earthquakes. This lesson goes over how to determine the inverse of a matrix and why it might be useful.

Matrix Inverse Explained

Olivia is one of those girls that loves computer games so much she wants to design them when she grows up. She has just learned that game graphics often make use of a powerful mathematical tool called matrices to make all that cool stuff appear on her screen. She wants to learn about these tools so she can get a leg up on her game design education.

Matrices, although cumbersome to use by hand, are very useful when employed by computers and can solve difficult problems very quickly - such as how a complicated digital monster might look as it is running quickly towards you. Olivia soon learns there are ways to add, subtract and multiply matrices, but there is no matrix operation equivalent of division. The closest we can get to division by a matrix is multiplying by its inverse.

Olivia knows from operations with integer numbers that dividing by a number gives you the same answer as multiplying by its reciprocal. 10 / 5 = 10 x (1/5) = 2. The same is true for the matrix inverses - as long as that matrix has an inverse. We'll see that not all matrices have an inverse.

Definition and Properties of the Inverse of a Matrix

Let's tighten up our loose definition of matrix inverses with some math:

Inverse Matrix properties

'What is an identity matrix?' Olivia wonders. She reads a little further and finds that the identity matrix has the same number of rows and columns, has '1' in every spot of the diagonal from upper left to lower right, and has '0' everywhere else. Can you see the pattern in the matrices below?

2x2 Identity Matrix
2x2 identity matrix

4x4 Identity Matrix
4x4 Identity

Identity matrices can be any size needed: 3x3, 10x10, even 1000x1000. Three dimensional computer graphics typically use 3x3 matrices, but apply them to tens of thousands of individual points that make up monsters, the landscape, and weapons you interact with on the screen.

There are a couple of properties to note about the inverse of a matrix. First, if you are multiplying a matrix by its inverse, the order does not matter. This is highly unusual for matrix operations because AB rarely equals BA for most matrices. Second, the inverse of a matrix might not even exist. When the determinant of a matrix is zero, you can't divide by that!

How to Calculate the Inverse of a 2x2 Matrix

To get the inverse of a 2x2 matrix, you need to take several steps:

  1. Switch the numbers in (row 1, column 1) and (row 2, column 2)
  2. Give opposite signs to the numbers in (row 1, column 2) and (row 2, column 1)
  3. Divide by the determinant of the original matrix

A visual aid is best here:

How to calculate inverse of 2x2 matrix

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