Determinants and Inverses

In the language of functions an inverse function say $f^{-1}(x)$ undoes whatever the function $f(x)$ does. I often make the analogy that a function is a set of instructions say to tie your shoes. The inverse function is the set of instructions to untie your shoes. A similar story is found with Matrices…

#### Matrix Inverses

A matrix, especially in multiplication, can be viewed as a set of instructions. For example below matrix A operates on matrix B to create matrix C.

(1)
\begin{equation} A B = C \end{equation}

So whatever operations A did to B might be able to be undone… This would be (un)done by the inverse of matrix A (because inverses undo) usually rewritten as $A^{-1}$. If the inverse exists then:

(2)
\begin{equation} A^{-1} A = I \end{equation}

Where I is the identity matrix of the appropriate size. This also lets us say, using the example above, that:

(3)
\begin{equation} A^{-1} A B = A^{-1} C \end{equation}

And thus that:

(4)
\begin{equation} B = A^{-1} C \end{equation}

A side note: The usefulness of inverses begin much more apparent when solving systems of equations...

#### 2 x 2 Inverse

There are several methods to find the inverse of a matrix (if it exists), but for the purposes of SL Math they are not enlightening, so we'll jump straight to the "good part." The inverse of the $2x2$ matrix A, where A is:

(5)
\begin{align} A=\begin{pmatrix} a & b\\ c & d \end{pmatrix} \end{align}

can be found with the following equation:

(6)
\begin{align} A^{-1}= \frac {1}{ad-bc}\begin{pmatrix} d & -b\\ -c & a \end{pmatrix} \end{align}

This equation is given in the IB Data Booklet. Looking at the equation we can see that there is a problem if $ad-bc=0$. If that quantity does in fact equal zero then the inverse does not exist, and the original matrix is said to be "singular", since dividing by zero doesn't work…

#### Determinant for a 2 x 2

For the $2x2$ matrix:

(7)
\begin{align} A=\begin{pmatrix} a & b\\ c & d \end{pmatrix} \end{align}

The determinant for a $2x2$ is given by the equation:

(8)
\begin{equation} det A = ad-bc \end{equation}

This equation is given in the IB data booklet. The determinant of a matrix can also be written with vertical bars such as:

(9)
\begin{align} det A = |A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix} \end{align}

Note that this is the same term that showed up while calculating the inverse. Thus we can say (at least for a $2x2$) that if the $det A=0$ then the inverse does not exist.

#### Determinant for a 3 x 3

Given a $3x3$ matrix:

(10)
\begin{align} B = \begin{pmatrix} a & b & c\\ d & e & f\\ g & h & k \end{pmatrix} \end{align}

Then the determinant of the matrix is given by:

(11)
\begin{align} det B =a\begin{vmatrix} e & f\\ h & k \end{vmatrix} - b\begin{vmatrix} d & f\\ g & k \end{vmatrix} +c\begin{vmatrix} d & e\\ g & h \end{vmatrix} \end{align}

This equation is also given in your IB data booklet.

#### Inverse of a 3 x 3 or Bigger

In SL Math you are NOT required to be able to calculate the inverse of anything bigger than a $2x2$ by hand. For a $3x3$ or bigger you should use your GDC. The video below goes through the basic steps of how to enter a matrix and calculate the inverse:

#### Existence of a Determinant

Determinants only exist or are only defined for square matrices… If a square matrix has a determinant of zero then the matrix is said to be singular.

#### Existence of an Inverse

In order for an inverse of a matrix to exist the matrix must be square and the determinant must be non-zero. This is true no matter the order of the matrix.