# 7.5 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.

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

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
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