10.8 The inverse of the generic 2x2 matrix

Write the matrix equation that defines $A^{-1}$ as an inverse of $A$ .

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

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.

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

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

$A^{-1}=$

As the final step, to put this in the form that it is most commonly seen in, note that all four terms have an $\mathrm{ad}-\mathrm{bc}$ in the denominator. (*Do you have a $\mathrm{bc}-\mathrm{ad}$ 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 $\frac{1}{\text{ad}-\text{bc}}$ . (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.

$A^{-1}=$

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!

The matrix $\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$

• 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.

The matrix $\left[\begin{array}{cc}3& 2\\ 9& 5\end{array}\right]$

• 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.

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