# What is the meaning of the phrase invertible matrix?

Aug 21, 2014

The short answer is that in a system of linear equations if the coefficient matrix is invertible, then your solution is unique, that is, you have one solution.

There are many properties for an invertible matrix to list here, so you should look at the Invertible Matrix Theorem . For a matrix to be invertible, it must be square , that is, it has the same number of rows as columns.

In general, it is more important to know that a matrix is invertible, rather than actually producing an invertible matrix because it is more computationally expense to calculate the invertible matrix compared to just solving the system. You would compute an inverse matrix if you were solving for many solutions.

Suppose you have this system of linear equations:

$2 x + 1.25 y = {b}_{1}$
$2.5 x + 1.5 y = {b}_{2}$

and you need to solve $\left(x , y\right)$ for the pairs of constants: $\left(119.75 , 148\right) , \left(76.5 , 94.5\right) , \left(152.75 , 188.5\right)$. Looks like a lot of work! In matrix form, this system looks like:

$A x = b$

where $A$ is the coefficient matrix, $x$ is the vector $\left(x , y\right)$ and $b$ is the vector $\left({b}_{1} , {b}_{2}\right)$. We can solve for $x$ with some matrix algebra:

$x = {A}^{- 1} b$

where ${A}^{- 1}$ is the inverse matrix. There are different ways to compute the inverse matrix, so I won't go into that now.

${A}^{- 1} =$
$\left[- 12 , 10\right]$
$\left[20 , - 16\right]$

So to get the solutions, we have:

$- 12 \cdot 119.75 + 10 \cdot 148 = 43 = {x}_{1}$
$20 \cdot 119.75 - 16 \cdot 148 = 27 = {y}_{1}$

$- 12 \cdot 76.5 + 10 \cdot 94.5 = 27 = {x}_{2}$
$20 \cdot 76.5 - 16 \cdot 94.5 = 18 = {y}_{2}$

$- 12 \cdot 152.75 + 10 \cdot 188.5 = 52 = {x}_{3}$
$20 \cdot 152.75 - 16 \cdot 188.5 = 39 = {y}_{3}$

Now, isn't that easier than solving 3 systems?