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

 Page 2 / 10

## Adding and subtracting matrices

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.

## Adding and subtracting matrices

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.

Matrix addition is commutative.

$A+B=B+A$

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

Add corresponding entries.

## 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.
1. Add the corresponding entries.
$\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] .

#### Questions & Answers

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
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Dominic
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
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_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
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Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
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Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
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corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
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16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
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4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
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Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
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-1
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