# Sage: Basic Matrix Operations

For using Sage: Sage is an open source math software whose interface is a web browser (in particular firefox). You don’t have to install Sage in your computer to use it. You can access any sage server including the main Sage server. I am running a Sage server at sage.st.usm.edu. If you are a student at the University of Southern Mississippi, you are more than welcome to create an account at sage.st.usm.edu and use it.

Matrix Constructions

In Sage, $2\times 3$ matrix
$$\begin{pmatrix} 1 & 1 & -2\\ -1 & 4 & -5 \end{pmatrix}$$ can be created as follows. Let us say we want to call the matrix $A$. Type the following command in the blank line of your sage worksheet:

A=matrix([[1,1,-2],[-1,4,-5]])

In case you are familiar with Maple, not like Maple you will not see your matrix $A$ as an output when you click on “evaluate”. To see your matrix, you need to type

A

in the next blank line and click on “evaluate”again:

[ 1  1 -2]
[-1  4 -5]

Scalar Multiplication

If you want to multiply the matrix $A$ by a number 5, type the command

5*A

and click on “evaluate”. The output is

[  5   5 -10]
[ -5  20 -25]

To perform addition of two matrices:
$$\begin{pmatrix} 1 & 1 & -2\\ -1 & 4 & -5 \end{pmatrix}+\begin{pmatrix} 2 & 1 & 5\\ 1 & 3 & 2 \end{pmatrix}$$, first call the second matrix $B$:

B=matrix([[2,1,5],[1,3,2]])

and do

A+B

the output is

[ 3  2  3]
[ 0  7 -3]

The linear combination $3A+2B$ can be calculated by the command

3*A+2*B

and the output is

[  7   5   4]
[ -1  18 -11]

Transpose of a Matrix

To find the transpose of the matrix $A$ do

A.transpose()

and the output is

[ 1 -1]
[ 1  4]
[-2 -5]

Matrix Multiplication

An $m\times n$ matrix can be multiplied by a $p\times q$ matrix as long as $n=p$. The resulting multiplication is an $m\times q$ matrix. Let $C=\begin{pmatrix} 3 & 4\\ -1 &2\\ 2 &1 \end{pmatrix}$. The number of columns of $A$ and the number of rows of $C$ coincide as 3, so we can perform $AC$ and this can be done in Sage as:

A*C

The output is

[ 1 -1]
[ 1  4]
[-2 -5]