Matrix Addition and Scalar Multiplication

You add two matrices of the same dimension by adding corresponding entries. The numbers or other expressions used as matrix entries are called scalars. You multiply a scalar with a matrix by multiplying every entry of the matrix by the scalar. You can do matrix addition and multiplication and other operations with scalars and matrices by choosing Evaluate. Place the insertion point anywhere inside the expression.

$\blacktriangleright$ Evaluate

$\left[\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right.$$\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}$$\left.\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right]$ + $\left[\vphantom{
\begin{array}{cc}
5 & 6 \\
8 & 7
\end{array}
}\right.$$\begin{array}{cc}
5 & 6 \\
8 & 7
\end{array}$$\left.\vphantom{
\begin{array}{cc}
5 & 6 \\
8 & 7
\end{array}
}\right]$ = $\left[\vphantom{
\begin{array}{cc}
6 & 8 \\
12 & 10
\end{array}
}\right.$$\begin{array}{cc}
6 & 8 \\
12 & 10
\end{array}$$\left.\vphantom{
\begin{array}{cc}
6 & 8 \\
12 & 10
\end{array}
}\right]$

Note that the sum appears with the same brackets as the original matrices.

$\blacktriangleright$ Evaluate

$\left(\vphantom{
\begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}
}\right.$$\begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}$$\left.\vphantom{
\begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}
}\right)$ +  $\left(\vphantom{
\begin{array}{cc}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{array}
}\right.$$\begin{array}{cc}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{array}$$\left.\vphantom{
\begin{array}{cc}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{array}
}\right)$ =  $\left(\vphantom{
\begin{array}{cc}
a_{11}+b_{11} & a_{12}+b_{12} \\
a_{21}+b_{21} & a_{22}+b_{22}
\end{array}
}\right.$$\begin{array}{cc}
a_{11}+b_{11} & a_{12}+b_{12} \\
a_{21}+b_{21} & a_{22}+b_{22}
\end{array}$$\left.\vphantom{
\begin{array}{cc}
a_{11}+b_{11} & a_{12}+b_{12} \\
a_{21}+b_{21} & a_{22}+b_{22}
\end{array}
}\right)$

a$\left[\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right.$$\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}$$\left.\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right]$ =  $\left[\vphantom{
\begin{array}{cc}
a & 2a \\
4a & 3a
\end{array}
}\right.$$\begin{array}{cc}
a & 2a \\
4a & 3a
\end{array}$$\left.\vphantom{
\begin{array}{cc}
a & 2a \\
4a & 3a
\end{array}
}\right]$

a$\left[\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right.$$\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}$$\left.\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right]$ - b$\left[\vphantom{
\begin{array}{cc}
5 & 6 \\
8 & 7
\end{array}
}\right.$$\begin{array}{cc}
5 & 6 \\
8 & 7
\end{array}$$\left.\vphantom{
\begin{array}{cc}
5 & 6 \\
8 & 7
\end{array}
}\right]$ =  $\left[\vphantom{
\begin{array}{cc}
a-5b & 2a-6b \\
4a-8b & 3a-7b
\end{array}
}\right.$$\begin{array}{cc}
a-5b & 2a-6b \\
4a-8b & 3a-7b
\end{array}$$\left.\vphantom{
\begin{array}{cc}
a-5b & 2a-6b \\
4a-8b & 3a-7b
\end{array}
}\right]$