10.6 The identity and inverse matrices

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This module provides sample problems which develop concepts related to the identity and inverse matrices.

This assignment is brought to you by one of my favorite numbers, and I’m sure it’s one of yours…the number 1. Some people say that 1 is the loneliest number that you’ll ever do. (*Bonus: who said that?) But I say, 1 is the multiplicative identity.

Allow me to demonstrate.

$5×1=$

$1×\frac{2}{3}=$

$–\pi ×1=$

$1×x=$

You get the idea? 1 is called the multiplicative identity because it has this lovely property that whenever you multiply it by anything, you get that same thing back. But that’s not all! Observe…

$2×\frac{1}{2}=$

$\frac{-2}{3}x\frac{-3}{2}=$

The fun never ends! The point of all that was that every number has an inverse. The inverse is defined by the fact that, when you multiply a number by its inverse, you get 1.

Write the equation that defines two numbers $a$ and $b$ as inverses of each other.

Find the inverse of $\frac{4}{5}$ .

Find the inverse of –3.

Find the inverse of $x$ .

Is there any number that does not have an inverse, according to your definition in #7?

So, what does all that have to do with matrices? (I hear you cry.) Well, we’ve already seen a matrix which acts as a multiplicative identity! Do these problems.

$\left[\begin{array}{cc}3& 8\\ -4& 12\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=$

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}3& 8\\ -4& 12\end{array}\right]=$

Pretty nifty, huh? When you multiply $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ by another 2×2 matrix, you get that other matrix back. That’s what makes this matrix (referred to as $\left[I\right]$ ) the multiplicative identity.

Remember that matrix multiplication does not, in general, commute: that is, for any two matrices $\left[A\right]$ and $\left[B\right]$ , the product $\mathrm{AB}$ is not necessarily the same as the product BA. But in this case, it is: $\left[I\right]$ times another matrix gives you that other matrix back no matter which order you do the multiplication in. This is a key part of the definition of $I$ , which is…

Definition of [i]

The matrix $I$ is defined as the multiplicative identity if it satisfies the equation: $\mathrm{AI}=\mathrm{IA}=A$

Which, of course, is just a fancy way of saying what I said before. If you multiply $I$ by any matrix, in either order, you get that other matrix back.

We have just seen that $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ acts as the multiplicative identify for a 2×2 matrix.

• A

What is the multiplicative identity for a 3×3 matrix?
• B

Test this identity to make sure it works.
• C

What is the multiplicative identity for a 5×5 matrix? (I won’t make you test this one…)
• D

What is the multiplicative identity for a 2×3 matrix?
• E

Trick question! There isn’t one. You could write a matrix that satisfies $\mathrm{AI}=A$ , but it would not also satisfy $\mathrm{IA}=A$ —that is, it would not commute, which we said was a requirement. Don’t take my word for it, try it! The point is that only square matrices (*same number of rows as columns) have an identity matrix.

So what about those inverses? Well, remember that two numbers $a$ and $b$ are inverses if $ab=1$ . As you might guess, we’re going to define two matrices $A$ and $B$ as inverses if $AB=\left[I\right]$ . Let’s try a few.

Multiply: $\left[\begin{array}{cc}2& 2\frac{1}{2}\\ -1& -1\frac{1}{2}\end{array}\right]$ $\left[\begin{array}{cc}3& 5\\ -2& -4\end{array}\right]$

Multiply: $\left[\begin{array}{cc}3& 5\\ -2& -4\end{array}\right]$ $\left[\begin{array}{cc}2& 2\frac{1}{2}\\ -1& -1\frac{1}{2}\end{array}\right]$

You see? These two matrices are inverses : no matter which order you multiply them in, you get $\left[I\right]$ . We will designate the inverse of a matrix as ${A}^{-1}$ which looks like an exponent but isn’t really, it just means inverse matrix—just as we used ${f}^{-1}$ to designate an inverse function. Which leads us to…

Definition of a-1

The matrix ${A}^{-1}$ is defined as the multiplicative inverse of $A$ if it satisfies the equation: ${A}^{-1}A=A{A}^{-1}=I$ (*where $I$ is the identity matrix)

Of course, only a square matrix can have an inverse, since only a square matrix can have an $I$ ! Now we know what an inverse matrix does , but how do you find one?

Find the inverse of the matrix $\left[\begin{array}{cc}3& 2\\ 5& 4\end{array}\right]$

• A

Since we don’t know the inverse yet, we will designate it as a bunch of unknowns: $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ will be our inverse matrix. Write down the equation that defines this unknown matrix as our inverse matrix.
• B

Now, in your equation, you had a matrix multiplication. Go ahead and do that multiplication, and write a new equation which just sets two matrices equal to each other.
• C

Now, remember that when we set two matrices equal to each other, every cell must be equal. So, when we set two different 2x2 matrices equal, we actually end up with four different equations. Write these four equations.
• D

Solve for $a$ , $b$ , $c$ , and $d$ .
• E

So, write the inverse matrix ${A}^{-1}$ .
• F

Test this inverse matrix to make sure it works!

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