# What is the meaning of the phrase invertible matrix?

##### 1 Answer

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:

#2x+1.25y=b_1#

#2.5x+1.5y=b_2#

and you need to solve

#Ax=b#

where

#x=A^(-1)b#

where

#A^(-1)=#

#[-12, 10]#

#[20, -16]#

So to get the solutions, we have:

#-12*119.75+10*148=43=x_1#

#20*119.75-16*148=27=y_1#

#-12*76.5+10*94.5=27=x_2#

#20*76.5-16*94.5=18=y_2#

#-12*152.75+10*188.5=52=x_3#

#20*152.75-16*188.5=39=y_3#

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