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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.
$5\times 1=$
$1\times \frac{2}{3}=$
$\mathrm{\u2013\pi}\times 1=$
$1\times x=$
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…
$2\times \frac{1}{2}=$
$\frac{-2}{3}x\frac{-3}{2}=$
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?
$\left[\begin{array}{cc}3& 8\\ -4& 12\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=$
$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}3& 8\\ -4& 12\end{array}\right]=$
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…
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.
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…
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 $\left[\begin{array}{cc}3& 2\\ 5& 4\end{array}\right]$
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