# 9.5 Matrices and matrix operations  (Page 5/10)

 Page 5 / 10

## Using a calculator to perform matrix operations

Find $\text{\hspace{0.17em}}AB-C\text{\hspace{0.17em}}$ given

On the matrix page of the calculator, we enter matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ above as the matrix variable $\text{\hspace{0.17em}}\left[A\right],$ matrix $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ above as the matrix variable $\text{\hspace{0.17em}}\left[B\right],$ and matrix $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ above as the matrix variable $\text{\hspace{0.17em}}\left[C\right].$

On the home screen of the calculator, we type in the problem and call up each matrix variable as needed.

$\left[A\right]×\left[B\right]-\left[C\right]$

The calculator gives us the following matrix.

$\left[\begin{array}{rrr}\hfill -983& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-462& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}136\\ \hfill 1,820& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}1,897& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-856\\ \hfill -311& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}2,032& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}413\end{array}\right]$

Access these online resources for additional instruction and practice with matrices and matrix operations.

## Key concepts

• A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.
• The dimensions of a matrix refer to the number of rows and the number of columns. A $\text{\hspace{0.17em}}3×2\text{\hspace{0.17em}}$ matrix has three rows and two columns. See [link] .
• Scalar multiplication involves multiplying each entry in a matrix by a constant. See [link] .
• Scalar multiplication is often required before addition or subtraction can occur. See [link] .
• Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must match the number of rows in the second.
• The product of two matrices, $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B,$ is obtained by multiplying each entry in row 1 of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by each entry in column 1 of $\text{\hspace{0.17em}}B;\text{\hspace{0.17em}}$ then multiply each entry of row 1 of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by each entry in columns 2 of $\text{\hspace{0.17em}}B,\text{}$ and so on. See [link] and [link] .
• Many real-world problems can often be solved using matrices. See [link] .
• We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. See [link] .

## Verbal

Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.

No, they must have the same dimensions. An example would include two matrices of different dimensions. One cannot add the following two matrices because the first is a $\text{\hspace{0.17em}}2×2\text{\hspace{0.17em}}$ matrix and the second is a $\text{\hspace{0.17em}}2×3\text{\hspace{0.17em}}$ matrix. $\text{\hspace{0.17em}}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{ccc}6& 5& 4\\ 3& 2& 1\end{array}\right]\text{\hspace{0.17em}}$ has no sum.

Can we multiply any column matrix by any row matrix? Explain why or why not.

Can both the products $\text{\hspace{0.17em}}AB\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}BA\text{\hspace{0.17em}}$ be defined? If so, explain how; if not, explain why.

Yes, if the dimensions of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}m×n\text{\hspace{0.17em}}$ and the dimensions of $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}n×m,\text{}$ both products will be defined.

Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.

Does matrix multiplication commute? That is, does $\text{\hspace{0.17em}}AB=BA?\text{\hspace{0.17em}}$ If so, prove why it does. If not, explain why it does not.

Not necessarily. To find $\text{\hspace{0.17em}}AB,\text{}$ we multiply the first row of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by the first column of $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ to get the first entry of $\text{\hspace{0.17em}}AB.\text{\hspace{0.17em}}$ To find $\text{\hspace{0.17em}}BA,\text{}$ we multiply the first row of $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ by the first column of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ to get the first entry of $\text{\hspace{0.17em}}BA.\text{\hspace{0.17em}}$ Thus, if those are unequal, then the matrix multiplication does not commute.

## Algebraic

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

$A=\left[\begin{array}{cc}1& 3\\ 0& 7\end{array}\right],B=\left[\begin{array}{cc}2& 14\\ 22& 6\end{array}\right],C=\left[\begin{array}{cc}1& 5\\ 8& 92\\ 12& 6\end{array}\right],D=\left[\begin{array}{cc}10& 14\\ 7& 2\\ 5& 61\end{array}\right],E=\left[\begin{array}{cc}6& 12\\ 14& 5\end{array}\right],F=\left[\begin{array}{cc}0& 9\\ 78& 17\\ 15& 4\end{array}\right]$

$A+B$

$C+D$

$\left[\begin{array}{cc}11& 19\\ 15& 94\\ 17& 67\end{array}\right]$

$A+C$

$B-E$

$\left[\begin{array}{cc}-4& 2\\ 8& 1\end{array}\right]$

$C+F$

$D-B$

Undidentified; dimensions do not match

For the following exercises, use the matrices below to perform scalar multiplication.

$A=\left[\begin{array}{rr}\hfill 4& \hfill 6\\ \hfill 13& \hfill 12\end{array}\right],B=\left[\begin{array}{rr}\hfill 3& \hfill 9\\ \hfill 21& \hfill 12\\ \hfill 0& \hfill 64\end{array}\right],C=\left[\begin{array}{rrrr}\hfill 16& \hfill 3& \hfill 7& \hfill 18\\ \hfill 90& \hfill 5& \hfill 3& \hfill 29\end{array}\right],D=\left[\begin{array}{rrr}\hfill 18& \hfill 12& \hfill 13\\ \hfill 8& \hfill 14& \hfill 6\\ \hfill 7& \hfill 4& \hfill 21\end{array}\right]$

$5A$

$3B$

$\left[\begin{array}{cc}9& 27\\ 63& 36\\ 0& 192\end{array}\right]$

$-2B$

$-4C$

$\left[\begin{array}{cccc}-64& -12& -28& -72\\ -360& -20& -12& -116\end{array}\right]$

$\frac{1}{2}C$

$100D$

$\left[\begin{array}{ccc}1,800& 1,200& 1,300\\ 800& 1,400& 600\\ 700& 400& 2,100\end{array}\right]$

For the following exercises, use the matrices below to perform matrix multiplication.

$A=\left[\begin{array}{rr}\hfill -1& \hfill 5\\ \hfill 3& \hfill 2\end{array}\right],B=\left[\begin{array}{rrr}\hfill 3& \hfill 6& \hfill 4\\ \hfill -8& \hfill 0& \hfill 12\end{array}\right],C=\left[\begin{array}{rr}\hfill 4& \hfill 10\\ \hfill -2& \hfill 6\\ \hfill 5& \hfill 9\end{array}\right],D=\left[\begin{array}{rrr}\hfill 2& \hfill -3& \hfill 12\\ \hfill 9& \hfill 3& \hfill 1\\ \hfill 0& \hfill 8& \hfill -10\end{array}\right]$

$AB$

$BC$

$\left[\begin{array}{cc}20& 102\\ 28& 28\end{array}\right]$

$CA$

$BD$

$\left[\begin{array}{ccc}60& 41& 2\\ -16& 120& -216\end{array}\right]$

$DC$

$CB$

$\left[\begin{array}{ccc}-68& 24& 136\\ -54& -12& 64\\ -57& 30& 128\end{array}\right]$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.

$A=\left[\begin{array}{rr}\hfill 2& \hfill -5\\ \hfill 6& \hfill 7\end{array}\right],B=\left[\begin{array}{rr}\hfill -9& \hfill 6\\ \hfill -4& \hfill 2\end{array}\right],C=\left[\begin{array}{rr}\hfill 0& \hfill 9\\ \hfill 7& \hfill 1\end{array}\right],D=\left[\begin{array}{rrr}\hfill -8& \hfill 7& \hfill -5\\ \hfill 4& \hfill 3& \hfill 2\\ \hfill 0& \hfill 9& \hfill 2\end{array}\right],E=\left[\begin{array}{rrr}\hfill 4& \hfill 5& \hfill 3\\ \hfill 7& \hfill -6& \hfill -5\\ \hfill 1& \hfill 0& \hfill 9\end{array}\right]$

$A+B-C$

$4A+5D$

Undefined; dimensions do not match.

$2C+B$

$3D+4E$

$\left[\begin{array}{ccc}-8& 41& -3\\ 40& -15& -14\\ 4& 27& 42\end{array}\right]$

$C-0.5D$

$100D-10E$

$\left[\begin{array}{ccc}-840& 650& -530\\ 330& 360& 250\\ -10& 900& 110\end{array}\right]$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: $\text{\hspace{0.17em}}{A}^{2}=A\cdot A$ )

$A=\left[\begin{array}{rr}\hfill -10& \hfill 20\\ \hfill 5& \hfill 25\end{array}\right],B=\left[\begin{array}{rr}\hfill 40& \hfill 10\\ \hfill -20& \hfill 30\end{array}\right],C=\left[\begin{array}{rr}\hfill -1& \hfill 0\\ \hfill 0& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right]$

$AB$

$BA$

$\left[\begin{array}{cc}-350& 1,050\\ 350& 350\end{array}\right]$

$CA$

$BC$

Undefined; inner dimensions do not match.

${A}^{2}$

${B}^{2}$

$\left[\begin{array}{cc}1,400& 700\\ -1,400& 700\end{array}\right]$

${C}^{2}$

${B}^{2}{A}^{2}$

$\left[\begin{array}{cc}332,500& 927,500\\ -227,500& 87,500\end{array}\right]$

${A}^{2}{B}^{2}$

${\left(AB\right)}^{2}$

$\left[\begin{array}{cc}490,000& 0\\ 0& 490,000\end{array}\right]$

${\left(BA\right)}^{2}$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: $\text{\hspace{0.17em}}{A}^{2}=A\cdot A$ )

$A=\left[\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 2& \hfill 3\end{array}\right],B=\left[\begin{array}{rrr}\hfill -2& \hfill 3& \hfill 4\\ \hfill -1& \hfill 1& \hfill -5\end{array}\right],C=\left[\begin{array}{rr}\hfill 0.5& \hfill 0.1\\ \hfill 1& \hfill 0.2\\ \hfill -0.5& \hfill 0.3\end{array}\right],D=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill -1\\ \hfill -6& \hfill 7& \hfill 5\\ \hfill 4& \hfill 2& \hfill 1\end{array}\right]$

$AB$

$\left[\begin{array}{ccc}-2& 3& 4\\ -7& 9& -7\end{array}\right]$

$BA$

$BD$

$\left[\begin{array}{ccc}-4& 29& 21\\ -27& -3& 1\end{array}\right]$

$DC$

${D}^{2}$

$\left[\begin{array}{ccc}-3& -2& -2\\ -28& 59& 46\\ -4& 16& 7\end{array}\right]$

${A}^{2}$

${D}^{3}$

$\left[\begin{array}{ccc}1& -18& -9\\ -198& 505& 369\\ -72& 126& 91\end{array}\right]$

$\left(AB\right)C$

$A\left(BC\right)$

$\left[\begin{array}{cc}0& 1.6\\ 9& -1\end{array}\right]$

## Technology

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.

$A=\left[\begin{array}{rrr}\hfill -2& \hfill 0& \hfill 9\\ \hfill 1& \hfill 8& \hfill -3\\ \hfill 0.5& \hfill 4& \hfill 5\end{array}\right],B=\left[\begin{array}{rrr}\hfill 0.5& \hfill 3& \hfill 0\\ \hfill -4& \hfill 1& \hfill 6\\ \hfill 8& \hfill 7& \hfill 2\end{array}\right],C=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 1& \hfill 0& \hfill 1\end{array}\right]$

$AB$

$BA$

$\left[\begin{array}{ccc}2& 24& -4.5\\ 12& 32& -9\\ -8& 64& 61\end{array}\right]$

$CA$

$BC$

$\left[\begin{array}{ccc}0.5& 3& 0.5\\ 2& 1& 2\\ 10& 7& 10\end{array}\right]$

$ABC$

## Extensions

For the following exercises, use the matrix below to perform the indicated operation on the given matrix.

$B=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 0\end{array}\right]$

${B}^{2}$

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

${B}^{3}$

${B}^{4}$

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

${B}^{5}$

Using the above questions, find a formula for $\text{\hspace{0.17em}}{B}^{n}.\text{\hspace{0.17em}}$ Test the formula for $\text{\hspace{0.17em}}{B}^{201}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{B}^{202},\text{}$ using a calculator.

${B}^{n}=\left\{\begin{array}{l}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\text{ }n\text{\hspace{0.17em}}\text{even,}\\ \left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right],\text{ }n\text{\hspace{0.17em}}\text{odd}\text{.}\end{array}$

How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture