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The Inverse of the Generic 2x2 Matrix

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides sample problems which develop concepts related to the generic 2x2 inverse matrix.

Today you are going to find the inverse of the generic 2×2 matrix. Once you have done that, you will have a formula that can be used to quickly find the inverse of any 2×2 matrix.

The generic 2×2 matrix, of course, looks like this:

[A] = abcd[A]=abcd size 12{ left [ matrix { a {} # b {} ## c {} # d{} } right ]} {}

Since its inverse is unknown, we will designate the inverse like this:

[ A-1 ] = wxyz[A-1]=wxyz size 12{ left [ matrix { w {} # x {} ## y {} # z{} } right ]} {}

Our goal is to find a formula for w w in terms of our original variables a a, b b, c c, and d d. That formula must not have any w w, x x, y y, or z z in it, since those are unknowns! Just the original four variables in our original matrix [A][A]. Then we will find similar formulae for x x, y y, and z z and we will be done.

Our approach will be the same approach we have been using to find an inverse matrix. I will walk you through the steps—after each step, you may want to check to make sure you’ve gotten it right before proceeding to the next.

Exercise 1

Write the matrix equation that defines A-1 A-1 as an inverse of AA.

Exercise 2

Now, do the multiplication, so you are setting two matrices equal to each other.

Exercise 3

Now, we have two 2×2 matrices set equal to each other. That means every cell must be identical, so we get four different equations. Write down the four equations.

Exercise 4

Solve. Remember that your goal is to find four equations—one for w w, one for x x, one for y y, and one for z—where each equation has only the four original constants a a, b b, c c, and d d!

Exercise 5

Now that you have solved for all four variables, write the inverse matrix A -1 A -1 .

A -1 = A -1 =

Exercise 6

As the final step, to put this in the form that it is most commonly seen in, note that all four terms have an ad-bcad-bc in the denominator. (*Do you have a bc-adbc-ad instead? Multiply the top and bottom by –1!) This very important number is called the determinant and we will see a lot more of it after the next test. In the mean time, note that we can write our answer much more simply if we pull out the common factor of 1adbc1adbc size 12{ { {1} over { ital "ad" - ital "bc"} } } {}. (This is similar to “pulling out” a common term from a polynomial. Remember how we multiply a matrix by a constant? This is the same thing in reverse.) So rewrite the answer with that term pulled out.

A -1 = A -1 =

You’re done! You have found the generic formula for the inverse of any 2x2 matrix. Once you get the hang of it, you can use this formula to find the inverse of any 2x2 matrix very quickly. Let’s try a few!

Exercise 7

The matrix 23452345 size 12{ left [ matrix { 2 {} # 3 {} ## 4 {} # 5{} } right ]} {}

  • a. Find the inverse—not the long way, but just by plugging into the formula you found above.
  • b. Test the inverse to make sure it works.

Exercise 8

The matrix 32953295 size 12{ left [ matrix { 3 {} # 2 {} ## 9 {} # 5{} } right ]} {}

  • a. Find the inverse—not the long way, but just by plugging into the formula you found above.
  • b. Test the inverse to make sure it works.

Exercise 9

Can you write a 2×2 matrix that has no inverse?

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