# 11.5 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.

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

answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
what is a algebra
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
cosA\1+sinA=secA-tanA
Wrong question
why two x + seven is equal to nineteen.
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI