**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]

**Matrix Addition**

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]