Build a Matrix from rows: 

>

 

Build a Matrix from columns: 

>

 

Multiply two matrices, and observe that matrix multiplication is not commutative: 

>

 

>

 

Find the inverse of the 2x2 matrix AB: 

>

 

Check that the inverse is correct, by hand:
(Note that the fractions were entered by typing, for example, 

"77/54" and Maple automatically typesets it the way you see.) 

>

 

Let's generate a random matrix, and see if it has an inverse: 

>

 

>

 

Since it has an inverse, it's reduced row-echelon form should be the identity matrix. 

>

 

Now let's see if the vector (20, 0, 0) is in the span of (1, 0, 2) and (2, 0, 3): 

Note the second way we do this, using LinearSolve.  This function assumes 

that we are trying to write the last column as a linear combination of the other 

columns, and gives us the coefficients in that linear combination. 

>

 

>

 

>

 

This last line tells us that (20, 0, 0) = -60(1, 0, 2) + 40(2, 0, 3).