Resources
What is a matrix inverse?
%Q OK. Now we can add, subtract and multiply matrices. That leaves one more operation: division. Specifically, if $A$ and $B$ are, say, $n \times n$ matrices, what is $\frac{A}{B}?$%A First ask yourself what division in the realm of real numbers actually means: it is really a form of multiplication: dividing $3$ by $7$ is the same as multiplying $3$ by $\frac{1}{7}$, the (multiplicative) inverse of $7.$ In other words,
$\frac{3}{7}$ \t $= 3\times\frac{1}{7}$
\\ \t $= 3\times7^{1}.$
Thus, in the realm of real numbers, we could actually forget all about division and just multiply by the inverse whenever we wanted to divide.
%Q Why complicate things like this? Just tell us how to divide matrices and be done with it!%A Patience! Writing down, say, $\frac{3}{7}$ instead of $3\cdot7^{1}$, is fine (and also customary) for real numbers. But should $\frac{3}{7}$ really mean $3\cdot7^{1}$ or should it mean $7^{1}\cdot 3$? Of course, it doesn't really matter, as multiplcation of real numbers is commutative. But multiplication of matrices is not commutative:
 By $\frac{A}{B}$ should we mean $A^{1}B$ or $BA^{1}$?
%A Not so fast! Before we try to find the inverse of a matrix $A,$ we must first know exactly what we mean by the (multiplicative) inverse. The inverse of a number $a$ is the number, often written $a^{1},$ with the property that
 $a^{1}a = 1 \quad$ %and $\quad a^{1}a = 1$
 $A^{1}A = 1 \quad$ %and $\quad A^{1}A = I \qquad \qquad$ The identity matrix $I$ plays the role of the number 1 in the realm of matrices.
Inverse of a matrix, Singular matrix
The* inverse of an $n \times n$ matrix $A$ is the $n \times n$ matrix $A^{1}$ such that
 $A^{1}A = 1 \quad$ %and $\quad A^{1}A = I \qquad \qquad$
Examples

The inverse of the $1 \times 1$ matrix
[7] %is[\sfrac{1}{7}] , %bc[7] [\sfrac{1}{7}] $=$[1] $= I\quad$ %and[\sfrac{1}{7}] [7] $=$[1] $= I.\quad$[7]^{1} $=$[\sfrac{1}{7}] and also[\sfrac{1}{7}]^{1} $=$[7] . 
The inverse of the $n \times n$ identity matrix $I$ is $I$ itself, because
 $I\cdot I = I.$

$I^{1} = I.$

[5,2;2,1]^1 $=$[1,2;2,5] because[5,2;2,1] [1,2;2,5] =[1,0;0,1] %and[1,2;2,5] [5,2;2,1] =[1,0;0,1] .
Finding the inverse of a matrix
%Q How on earth do we find the inverse of a matrix?%A If you used unknowns for all the entries of the inverse of a matrix, the condition that $AA^{1} = I$ would give you a system of linear equations. Solving this system of equations corresponds to the following method for finding the inverse of any matrix: (See %4 for a detailed explanation). %Let $A = $%10. Determine whether or not $A$ is invertible, and its inverse if it is, as follows: Step 1. Write down the $n\times 2n$ matrix
Using matrix inverses to solve systems of linear equations
Look at the following system of linear equations:
$x + y$ \t $=4$
\\ $4x + y + z$ \t $=1$
\\ $2y + z$ \t $=0$
To solve the system, we would normally write out and rowreduce the augmented matrix for the system.
However, there is another way to solve certain systems of equations like this one: Take $A$ to be the matrix of coefficients of the lefthand sides:

$A = $
 $AX=$
 $AX = B,$

$A^{1}AX = A^{1}B \qquad \Rightarrow \qquad IX = A^{1}B \qquad \Rightarrow \qquad X = A^{1}B,$
 $X = A^{1}B = $
Now try the exercises in %4, some the %8, or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.
Last Updated: January, 2017
Copyright © 2017
Copyright © 2017