# Simple Operations with Matrices

## (Addition, Subtraction & Multiplication by a Scalar)

In this section we learn about addition, subtraction, and multiplication by a scalar with matrices.

When adding and subtracting with matrices, the following important rule should always be kept in mind:

Only matrices that are of the same order can be added to, or subtracted from, each other.

So, for example, given the matrices $$A = \begin{pmatrix} 1 & -2 \\ 0 & 3\end{pmatrix}$$, $$B = \begin{pmatrix} 4 & 1 \\ 2 & -5 \end{pmatrix}$$ and $$C = \begin{pmatrix} 1 & -2 & 7\\ 0 & 3 & 5\end{pmatrix}$$ we can calculate: $$A+B$$, $$A-B$$ and $$B-A$$ but we cannot calculate $$A+C$$, $$B-C$$, ... since $$C$$ is of order $$\begin{pmatrix} 2\times 3\end{pmatrix}$$ and both $$A$$ and $$B$$ are of order $$2\times 2$$.

### Addition & Subtraction with Matrices

Given two matrices $$A$$ and $$B$$ of the same order, $$A = \begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$$ and $$B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}$$, we can add them together by adding corresponding entries together. Similarly, to subtract $$B$$ from $$A$$ we subtract each entry of $$B$$ from its corresponding entry in $$A$$.

This is shown here for $$2 \times 2$$ matrices. The method would be the same for any two matrices of the same order.

$A+B = \begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} + \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix}a_{11} + b_{11}& a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{pmatrix}$

#### Subtraction

$A - B = \begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} - \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix}a_{11} - b_{11}& a_{12} - b_{12} \\ a_{21} - b_{21} & a_{22} - b_{22} \end{pmatrix}$

## Example 1

Given the matrices $$A = \begin{pmatrix} 2 & 3 \\ 1 & -5 \end{pmatrix}$$ and $$B = \begin{pmatrix} -2 & 0 \\ 7 & -1 \end{pmatrix}$$, below we show how to calculate $$A+B$$ as well as $$A - b$$.

We find $$A+B$$ as follows: \begin{aligned} A + B &= \begin{pmatrix} 2 & 3 \\ 1 & -5 \end{pmatrix} + \begin{pmatrix} -2 & 0 \\ 7 & -1 \end{pmatrix} \\ & = \begin{pmatrix} 2 + (-2) & 3 + 0 \\ 1 + 7 & -5 + (-1) \end{pmatrix}\\ A + B & = \begin{pmatrix} 0 & 3 \\ 8 & -6 \end{pmatrix} \end{aligned}

#### Subtraction

We find $$A-B$$ as follows: \begin{aligned} A - B & = \begin{pmatrix} 2 & 3 \\ 1 & -5 \end{pmatrix} - \begin{pmatrix} -2 & 0 \\ 7 & -1 \end{pmatrix} \\ & = \begin{pmatrix} 2 + (-2) & 3 + 0 \\ 1 + 7 & -5 + (-1) \end{pmatrix}\\ A - B & = \begin{pmatrix} 0 & 3 \\ 8 & -6 \end{pmatrix} \end{aligned}

## Example 2

Given the matrices $$M = \begin{pmatrix} 3 & 5 \\ -2 & 0 \\ 1 & 6 \end{pmatrix}$$ and $$N = \begin{pmatrix} 0 & 2 \\ 3 & 4 \\ -7 & 1 \end{pmatrix}$$, below we show how to calculate $$M+N$$ as well as $$M - N$$.

\begin{aligned} M + N & = \begin{pmatrix} 3 & 5 \\ -2 & 0 \\ 1 & 6 \end{pmatrix} + \begin{pmatrix} 0 & 2 \\ 3 & 4 \\ -7 & 1 \end{pmatrix}\\ & = \begin{pmatrix} 3 +0 & 5 +2\\ -2 +3 & 0 + 4\\ 1 + (-7) & 6 +1\end{pmatrix} \\ M + N & = \begin{pmatrix} 3 & 7 \\ 1 & 4\\ -6 & 7 \end{pmatrix} \end{aligned}

#### Subtraction

\begin{aligned} M - N & = \begin{pmatrix} 3 & 5 \\ -2 & 0 \\ 1 & 6 \end{pmatrix} - \begin{pmatrix} 0 & 2 \\ 3 & 4 \\ -7 & 1 \end{pmatrix}\\ & = \begin{pmatrix} 3 - 0 & 5 - 2\\ -2 - 3 & 0 - 4\\ 1 - (-7) & 6 - 1\end{pmatrix} \\ M - N & = \begin{pmatrix} 3 & 3 \\ -5 & -4\\ 8 & 5 \end{pmatrix} \end{aligned}

### Multiplication by a Scalar

Given a matrix $$A$$, we can multiply $$A$$ by any scalar $$k \in \mathbb{R}$$ by multiplying each of the entries of $$A$$ by $$k$$. This is shown here for a $$2\times 2$$ matrix: $k\times A = k\times \begin{pmatrix} a_{11} & a_{11} \\ a_{11} & a_{11} \end{pmatrix} = \begin{pmatrix} k\times a_{11} & k\times a_{11} \\ k\times a_{11} & k\times a_{11} \end{pmatrix}$

## Example 3

1. Given the matrix $$A$$ defined as $$A = \begin{pmatrix} 1 & -3 \\ 0 & 5 \end{pmatrix}$$, we can multiply this matrix by $$2$$ as follows: \begin{aligned} 2\times A & = 2\times \begin{pmatrix} 1 & -3 \\ 0 & 5 \end{pmatrix} \\ & = \begin{pmatrix} 2\times 1 & 2\times (-3) \\ 2\times 0 & 2\times 5 \end{pmatrix} \\ 2A & = \begin{pmatrix} 2 & -6 \\ 0 & 10 \end{pmatrix} \end{aligned}

2. Given the matrix $$B$$ defined as $$B = \begin{pmatrix} 2 & 1 \\ -4 & 8 \\ 6 & 0 \end{pmatrix}$$, we can multiply this matrix by $$-3$$ as follows: \begin{aligned} -3 \times B & = -3\times \begin{pmatrix} 2 & 1 \\ -4 & 8 \\ 6 & 0 \end{pmatrix} \\ & = \begin{pmatrix} -3 \times 2 & -3 \times 1 \\ -3 \times (-4) & -3 \times 8 \\ -3 \times 6 & -3 \times 0 \end{pmatrix} \\ -3B & = \begin{pmatrix} -6 & -3 \\ 12 & -24 \\ -18 & 0 \end{pmatrix} \end{aligned}

## Linear Combinations

Given two matrices $$A$$ and $$B$$ of the same order and two scalars $$m$$ and $$n$$ in $$\mathbb{R}$$, we call a linear combination of $$A$$ and $$B$$ any expression that can be written: $m .A + n . B$

## Example 4

Given $$A = \begin{pmatrix} 1 & 0 & 3 \\ 2 & 7 & -4 \end{pmatrix}$$ and $$B = \begin{pmatrix} 3 & 2 & 5 \\ 1 & 6 & 9 \end{pmatrix}$$ find $$3A-2B$$.

#### Solution

\begin{aligned} 3A - 2B & = 3 \times \begin{pmatrix} 1 & 0 & 3 \\ 2 & 7 & -4 \end{pmatrix} - 2 \times \begin{pmatrix} 3 & 2 & 5 \\ 1 & 6 & 9 \end{pmatrix} \\ & = \begin{pmatrix} 3 & 0 & 9 \\ 6 & 21 & -12 \end{pmatrix} - \begin{pmatrix} 6 & 4 & 10 \\ 2 & 12 & 18 \end{pmatrix} \\ 3A - 2B & = \begin{pmatrix} -3 & -4 & -1 \\ 4 & 9 & -30 \end{pmatrix} \end{aligned}

## Exercise 1

1. Given the matrices $$A = \begin{pmatrix} 2 & 1 \\ 5 & -3 \\ 0 & 4\end{pmatrix}$$ and $$B = \begin{pmatrix} 6 & -1 \\ 2 & 0 \\ 1 & 3 \end{pmatrix}$$, find:
1. $$3A$$
2. $$-2B$$
3. $$-A$$
1. $$A+B$$
2. $$A - B$$
3. $$B-A$$

2. Given the matrices $$A = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix}$$, $$B = \begin{pmatrix} 5 & 0 \\ -6 & 2 \end{pmatrix}$$, $$C = \begin{pmatrix} 3 & -2 & 0 \\ 5 & 1 & 6 \\ 8 & 7 & -4 \end{pmatrix}$$ and $$D = \begin{pmatrix} 1 & 0 & -1 \\ 2 & 5 & 8 \\ 0 & 4 & 7 \end{pmatrix}$$, find:
1. $$A+B$$
2. $$3B-A$$
1. $$2C-D$$
2. $$3D + 2C$$

3. Given $$M = \begin{pmatrix} 3 & 0 & 2 \\ -5 & 1 & 4 \end{pmatrix}$$, $$N = \begin{pmatrix} 4 & -3 & 1 \\ 6 & 0 & 2 \end{pmatrix}$$ and $$P = \begin{pmatrix} 0 & 1 & 0 \\ 5 & -2 & 3 \end{pmatrix}$$, find:
1. $$2M+3P$$
2. $$3N-M$$
1. $$M + 2N + P$$
2. $$3M - N + P$$

Note: this exercise can be downloaded as a worksheet to practice with:

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