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)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:

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

(3)And thus that:

(4)**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)can be found with the following equation:

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

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

(9)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)Then the determinant of the matrix is given by:

(11)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.

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