01Zmatrixinverse


 * The Identity Matrix **


 * The ** Identity Matrix  ** (**I**) is a __ square matrix __ {equal number of rows and columns}.
 * It has 1s in the diagonal from top left to bottom right {where the row number equals the column number} and 0s everywhere else.

For example, here is the 4 x 4 Identity Matrix. math I = \left[ \begin{matrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \\ \end{matrix} \right] math


 * Multiplying by the Identity Matrix (**I**) is like multiplying by the number 1. It does not change the thing you are multiplying.
 * In numbers, 1 × a = a
 * In matrices, ** I ** × A = A


 * Example 1 **

math \text{Let } \, A = \left[ \begin{matrix} 1&2 \\ 3&4 \\ \end{matrix} \right] math then {using using the rules for multiplying matrices} math \\ I \times A = \left[ \begin{matrix} 1&0 \\ 0&1 \\ \end{matrix} \right] \times \left[ \begin{matrix} 1&2 \\ 3&4 \\ \end{matrix} \right] \\ \\ . \qquad \; = \left[ \begin{matrix} 1 \times 1 + 3\times 0&1 \times 2 + 0 \times 4 \\ 0 \times 1 + 1 \times 3& 0\times 2+ 1 \times 4 \\ \end{matrix} \right] \\ \\ . \qquad \; = \left[ \begin{matrix} 1&2 \\ 3&4 \\ \end{matrix} \right] math


 * The Inverse of a 2 by 2 Matrix **


 * For any matrix, A, the ** Inverse Matrix  **, A –1 , is defined so that:
 * A × A –1 = **I**
 * (and)
 * A –1 × A = **I**


 * {remember that when multiplying matrices the order is very important}


 * This is equivalent in normal arithmetic to multiplying a number by its reciprocal: 3 × 3 –1 = 1


 * To find the Inverse Matrix, A –1, of a 2 by 2 matrix we first have to calculate the ** __determinant__ ** of A.

The Determinant of a 2 by 2 Matrix

For any 2 by 2 matrix: math A = \left[ \begin{matrix} a&b \\ c&d \\ \end{matrix} \right] math the ** determinant **, |A|, is given by: |A| = ad – bc


 * Example 2 **

math \text{Let } \, A = \left[ \begin{matrix} 1&2 \\ 3&4 \\ \end{matrix} \right] math then
 * A| ``=`` 1 × 4 – 2 × 3 ``=`` –2

Finding the determinant on the Classpad

Recall that you can enter a matrix on the Classpad by going to the virtual keyboard, then selecting the 2D tab and choosing the CALC section. Increase the size of the matrix by tapping the matrix symbol again.

To find the determinant, use the **det** function in the ACTION menu, MATRIX-CALCULATION submenu. For example, enter: math \textbf{det}( \left[ \begin{matrix} 1&2 \\ 3&4 \\ \end{matrix} \right] ) math

The Inverse of a 2 by 2 Matrix


 * The inverse of a matrix does not exist if the determinant is 0.
 * A matrix with no index is called a ** singular matrix  **.


 * We construct the Inverse from the original by swapping the values in the a and d positions and making the b and c values negative.
 * Then multiply the resulting matrix by the reciprocal of the determinant; (1 divided by |A| )

So, for any 2 by 2 matrix: math \\ A = \left[ \begin{matrix} a&b \\ c&d \\ \end{matrix} \right] \\. \\ \\ A^{-1} = \dfrac{1}{|A|} \left[ \begin{matrix} d&-b \\ -c&a \\ \end{matrix} \right] math


 * Example 3 **

math \text{Let } \, A = \left[ \begin{matrix} 1&2 \\ 3&4 \\ \end{matrix} \right] math from above, we know that |A| = –2 then math A^{-1} = \dfrac{1}{-2} \left[ \begin{matrix} 4&-2 \\ -3&1 \\ \end{matrix} \right] math or math A^{-1} = \left[ \begin{matrix} -2&1 \\ 1.5&-0.5 \\ \end{matrix} \right] math {sometimes it is easier to leave the inverse with the fraction out the front}

We can demonstrate this is the inverse by multiplying the two matrices together. The product should be the Identity Matrix. math A^{-1} \times A \;\; = - \dfrac{1}{2} \left[ \begin{matrix} 4&-2 \\ -3&1 \\ \end{matrix} \right] \times \left[ \begin{matrix} 1&2 \\ 3&4 \\ \end{matrix} \right] math math \\ . \qquad \qquad = - \dfrac{1}{2} \left[ \begin{matrix} 4-6&8-8 \\ -3+3&-6+4 \\ \end{matrix} \right] \\ \\ . \qquad \qquad = - \dfrac{1}{2} \left[ \begin{matrix} -2&0 \\ 0&-2 \\ \end{matrix} \right] \\ \\ . \qquad \qquad = \left[ \begin{matrix} 1&0 \\ 0&1 \\ \end{matrix} \right] \\ \\ . \qquad \qquad = I math

The Inverse of Larger Matrices


 * To find the Inverse of a 3 by 3 matrix by hand is much more complicated.


 * And to find the inverse of anything larger by hand is an exercise in torture.


 * Fortunately, technology is capable of doing that for us so we don't have to.

Finding the Inverse Matrix on the Classpad

Recall that you can enter a matrix on the Classpad by going to the virtual keyboard, then selecting the 2D tab and choosing the CALC section. Increase the size of the matrix by tapping the matrix symbol again.

To find the Inverse Matrix, enter the matrix then ^(–1) For example, enter: math \left[ \begin{matrix} 1&2 \\ 3&4 \\ \end{matrix} \right]^{-1} math

The Classpad can find the inverse of larger matrices just as easily.

What's the point?


 * The Inverse of a matrix is used when solving simultaneous equations using matrices.

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 * For another site that explains this idea, go here: MathsIsFun