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

 Page 2 / 10

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] .

Need help solving this problem (2/7)^-2
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_