# 11.5 Matrices and matrix operations  (Page 2/10)

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We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.

In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions . We can add or subtract a matrix and another matrix, but we cannot add or subtract a matrix and a matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

Given matrices $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ of like dimensions, addition and subtraction of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ will produce matrix $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ or
matrix $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ of the same dimension.

$A+B=B+A$

It is also associative.

$\left(A+B\right)+C=A+\left(B+C\right)$

## Finding the sum of matrices

Find the sum of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B,\text{}$ given

## Adding matrix A And matrix B

Find the sum of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B.$

Add corresponding entries. Add the entry in row 1, column 1, $\text{\hspace{0.17em}}{a}_{11},\text{}$ of matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ to the entry in row 1, column 1, $\text{\hspace{0.17em}}{b}_{11},$ of $\text{\hspace{0.17em}}B.\text{\hspace{0.17em}}$ Continue the pattern until all entries have been added.

## Finding the difference of two matrices

Find the difference of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B.$

We subtract the corresponding entries of each matrix.

## Finding the sum and difference of two 3 x 3 matrices

Given $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B:$

1. Find the sum.
2. Find the difference.
$\begin{array}{l}\hfill \\ A+B=\left[\begin{array}{rrr}\hfill 2& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-10& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\\ \hfill 14& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}12& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}10\\ \hfill 4& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}2\end{array}\right]+\left[\begin{array}{rrr}\hfill 6& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}10& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\\ \hfill 0& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-12& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-4\\ \hfill -5& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}2& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\end{array}\right]\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left[\begin{array}{rrr}\hfill 2+6& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-10+10& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2-2\\ \hfill 14+0& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}12-12& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}10-4\\ \hfill 4-5& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2+2& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}2-2\end{array}\right]\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left[\begin{array}{rrr}\hfill 8& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}0& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-4\\ \hfill 14& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}0& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}6\\ \hfill -1& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}0& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}\right]\hfill \end{array}$
2. Subtract the corresponding entries.
$\begin{array}{l}\hfill \\ A-B=\left[\begin{array}{rrr}\hfill 2& \hfill -10& \hfill -2\\ \hfill 14& \hfill 12& \hfill 10\\ \hfill 4& \hfill -2& \hfill 2\end{array}\right]-\left[\begin{array}{rrr}\hfill 6& \hfill 10& \hfill -2\\ \hfill 0& \hfill -12& \hfill -4\\ \hfill -5& \hfill 2& \hfill -2\end{array}\right]\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left[\begin{array}{rrr}\hfill 2-6& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-10-10& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2+2\\ \hfill 14-0& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}12+12& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}10+4\\ \hfill 4+5& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2-2& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}2+2\end{array}\right]\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left[\begin{array}{rrr}\hfill -4& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-20& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ \hfill 14& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}24& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}14\\ \hfill 9& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-4& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}4\end{array}\right]\hfill \end{array}$

Add matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and matrix $\text{\hspace{0.17em}}B.$

$A+B=\left[\begin{array}{c}2\\ 1\\ 1\end{array}\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}6\\ \text{​}\text{​}\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-3\end{array}\right]+\left[\text{\hspace{0.17em}}\begin{array}{c}\text{\hspace{0.17em}}3\\ \text{\hspace{0.17em}}1\\ -4\end{array}\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}5\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3\end{array}\right]=\left[\begin{array}{c}2\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}3\\ 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\\ 1+\left(-4\right)\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}6+\left(-2\right)\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}5\\ -3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3\end{array}\right]=\left[\begin{array}{c}\text{\hspace{0.17em}}5\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}2\\ -3\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}4\\ 5\\ 0\end{array}\right]$

## Finding scalar multiples of a matrix

Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar    is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication    involves multiplying each entry in a matrix by a scalar. A scalar multiple    is any entry of a matrix that results from scalar multiplication.

Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school’s current inventory is displayed in [link] .

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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