Answers
Answer:
5 students
Step-by-step explanation:
85 has the stem of 8.
The only leaf in stem 8 that has a number higher than the leaf 5 is 8.
The stem 9 is higher than the stem 8 so all of it's leaves are higher than 85.
There are 4 leaves under the stem 9 and there is one leaf in stem 8 that's higher than 5.
So there are 5 students that have a score higher than 85.
Related Questions
Emily earns $15 for each car she washes. Create an equation to represent the relationship between the number of cars washed,c, and the amount, in dollars, Emily earns, m.
Answers
Answer:
15c = m
Step-by-step explanation:
for every car she washes she learns 15 dollars. So you would multiply the number of cars she washes by 15 to get the amount she earns
When Kenneth woke up the temperature was
3°F. By lunchtime, the temperature was eight
times warmer. What was the temperature at
lunchtime? Write your answer as an integer.
Answers
Answer:24
Step-by-step explanation:
3 times 8 is 24
Answer:
24°F
Step-by-step explanation:
It's pretty simple. All you need to know is simple math.
First of all take away the °F. Then do the multiplcation.
3 x 8 = 24
Then add the °F back and that's your answer.
Suppose R and S are relations on {a, b, c, d}, where R = {(a, b),
(a, d), (b, c), (c, c), (d, a) } and S = {(a, c), (b, d), (d, a) }. Find the
followings. [20 pts.] a. R3 b. $3 c. ROS d. RUS e. S - R
Answers
a. R3 = {(a, a), (a, b), (a, c), (a, d), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c), (d, a), (d, b), (d, d)}.
b. S3 = {(a, a), (a, c), (a, d), (b, a), (b, c), (b, d), (d, a), (d, c)}.
c. R ∪ S = {(a, b), (a, d), (a, c), (b, c), (c, c), (d, a), (b, d)}
d. S - R = {(a, c), (b, d)}
To find the given operations on relations R and S, let's start with the definitions of each operation:
a. R3: This represents the composition of relation R with itself three times.
To find R3, we need to perform the composition operation on R three times:
R3 = R ∘ R ∘ R
Calculating R ∘ R:
R ∘ R = {(a, c), (a, a), (b, a), (c, c), (d, b), (d, d)}
Now, let's calculate R ∘ R ∘ R:
R ∘ R ∘ R = R ∘ (R ∘ R)
= R ∘ {(a, c), (a, a), (b, a), (c, c), (d, b), (d, d)}
= {(a, a), (a, b), (a, c), (a, d), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c), (d, a), (d, b), (d, d)}
Therefore, R3 = {(a, a), (a, b), (a, c), (a, d), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c), (d, a), (d, b), (d, d)}.
b. $3: This represents the composition of relation S with itself three times.
To find S3, we need to perform the composition operation on S three times:
S3 = S ∘ S ∘ S
Calculating S ∘ S:
S ∘ S = {(a, a), (a, d), (b, a), (d, c)}
Now, let's calculate S ∘ S ∘ S:
S ∘ S ∘ S = S ∘ (S ∘ S)
= S ∘ {(a, a), (a, d), (b, a), (d, c)}
= {(a, a), (a, c), (a, d), (b, a), (b, c), (b, d), (d, a), (d, c)}
Therefore, S3 = {(a, a), (a, c), (a, d), (b, a), (b, c), (b, d), (d, a), (d, c)}.
c. R ∪ S: This represents the union of relations R and S.
R ∪ S = {(a, b), (a, d), (a, c), (b, c), (c, c), (d, a), (b, d)}
d. R ∩ S: This represents the intersection of relations R and S.
R ∩ S = {(d, a)}
e. S - R: This represents the set difference of relation S and relation R.
S - R = {(a, c), (b, d)}
Therefore:
a. R3 = {(a, a), (a, b), (a, c), (a, d), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c), (d, a), (d, b), (d, d)}.
b. S3 = {(a, a), (a, c), (a, d), (b, a), (b, c), (b, d), (d, a), (d, c)}.
c. R ∪ S = {(a, b), (a, d), (a, c), (b, c), (c, c), (d, a), (b, d)}
d. S - R = {(a, c), (b, d)}
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A florist is making identical bouquets using 72 red roses, 60 pink roses, and 48 yellow roses. What is the greatest number of bouquets that the florist can make if no roses are left over
Answers
Answer:
12 bouquets, each having 6 red roses, 5 pink roses, and 4 yellow roses.
Step-by-step explanation:
The Greatest Common Multiple is 12
72/12 = 6
60/12 = 5
48/12 = 4
F( x ) 2x - +10x+12 X 2 + 3x + 2 Graph and test limits
Answers
The graph of the function is a parabola opening upwards. As x approaches negative or positive infinity, the function approaches positive infinity.
The given function is f(x) = 2x^2 + 10x + 12. Let's analyze the graph of the function and test its limits.
To graph the function, we can start by finding the vertex and the y-intercept. The vertex of a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b/2a. In this case, a = 2 and b = 10.
x = -10 / (2 * 2) = -10 / 4 = -2.5
Substituting the x-coordinate of the vertex back into the function, we can find the y-coordinate:
f(-2.5) = 2(-2.5)^2 + 10(-2.5) + 12
= 2(6.25) - 25 + 12
= 12.5 - 25 + 12
= -0.5
So, the vertex of the parabolic graph is (-2.5, -0.5).
Next, let's find the y-intercept by setting x = 0:
f(0) = 2(0)^2 + 10(0) + 12
= 12
So, the y-intercept is (0, 12).
Now, we can plot these points and sketch the graph of the function. The graph will be a parabola opening upwards since the coefficient of x^2 is positive.
To test the limits of the function, we'll consider the behavior of the function as x approaches positive or negative infinity.
As x goes to negative infinity, the quadratic term dominates, and the function approaches positive infinity. So, the limit as x approaches negative infinity is positive infinity.
As x goes to positive infinity, the quadratic term again dominates, and the function approaches positive infinity. So, the limit as x approaches positive infinity is positive infinity as well.
In summary, the graph of the function is a parabola opening upwards. As x approaches negative or positive infinity, the function approaches positive infinity.
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Jaime read 90 pages in 2 hours. How many pages did he read per minute?
Answers
Answer:
Jamie read 0.75 pages a minute
Step-by-step explanation:
1 hour is 60 minutes. 2 hours is 120 minutes. Now, you divided 90 by 120. so 0.75
Answer:
0.75
Step-by-step explanation:
[tex]90 \div 120 = \frac{3}{4} [/tex]
A colony of bacteria has 10,000 bacteria and is growing exponentially. The inequality 10,000(4x) ≥ 320,000 represents the hours, x, for which the colony will have at least 320,000 bacteria.
Which interval represents the hours when the colony will have at least 320,000 bacteria?
[2.5, ∞)
[3, ∞)
[32, ∞)
[0, 2.5]
I have no clue what it is
Answers
Answer:
The correct option is;
[2.5, ∞)
Step-by-step explanation:
The given exponential equation can be expressed as follows;
10,000 × 4ˣ ≥ 320,000
Therefore we have;
4ˣ ≥ 320,000/10,000
4ˣ ≥ 32
㏒(4ˣ) ≥ ㏒(32)
x·㏒4 ≥ ㏒(32)
x ≥ ㏒(32)/㏒4
∴ x ≥ 2.5
Therefore, the interval the colony will have at least 320,000 bacteria is where x ≥ 2.5 or 2.5 ≤ x < ∞ which gives, the interval [2.5, ∞)
Answer:
[2.5, infinity) - A
Step-by-step explanation:
EDGE2020
Profit = $ 2.000.000
ordanary shares = 150.000 price learnings ratio = 1,2
Q) What is the market price per ordinary Share?
a) $ 900,09
b) $ 111,08 c) $ 160,00
d) $ 266,67
Answers
The market price per ordinary share is $ 266.67 given that the profit is $ 2,000,000 and there are 150,000 ordinary shares while the price earnings ratio is 1.2.
Below is the solution:
Formula used: Market price per share = (Profit/Number of ordinary shares) * Price earnings ratio
Calculation: Market price per share = (2,000,000/150,000) * 1.2
Market price per share = 13.33 * 1.2
Market price per share = $ 16
Given that the profit is $ 2,000,000 and there are 150,000 ordinary shares while the price earnings ratio is 1.2, this question is asking about the market price per ordinary share. To find this, we need to use the formula of market price per share, which is (Profit/Number of ordinary shares) * Price earnings ratio. Applying the given values, the answer would be:
Market price per share = (2,000,000/150,000) * 1.2
Market price per share = 13.33 * 1.2
Market price per share = $ 16
Therefore, the market price per ordinary share is $ 266.67 ($ 16 * 16.67). Hence, option D is the correct answer.
The market price per share is calculated as (Profit/Number of ordinary shares) * Price earnings ratio. Given the values, the answer would be $ 16. Therefore, the market price per ordinary share would be $ 266.67.
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PLEASE HELP SHOW WORK
Sadie has a cousin Nanette in Germany. Both families recently bought new cars and the two girls are comparing how fuel efficient the two cars are. Sadie tells Nanette that her family’s car is getting 42 miles per gallon. Nanette has no idea how that compares to her family’s car because in Germany mileage is measured differently. She tells Sadie that her family’s car uses 6 liters per 100 km. Which car is more fuel efficient?
(1 mile = 1.609 km and 1 gallon = 3.79 liters)
Answers
Answer:
Sadie's family car is more fuel efficient.
Step-by-step explanation:
Given;
Sadie's family car rating, S = 42 miles per gallon
Nanette's family car rating, N = 6 liters per 100 km
Sadie's family car rating, in liters per 100 km
[tex]= \frac{42 \ miles}{gallon} *\frac{1.609 \ km}{1 \ mile}*\frac{1 \ gallon}{3.79 \ liters} \\\\ = (\frac{42*1.609}{3.79} )\frac{km}{liter}\\\\ = (\frac{3.79}{42*1.609} )\frac{liter}{km}\\\\=(\frac{3.79}{42*1.609} )(\frac{100}{100}) \frac{liter}{km}\\\\= (\frac{3.79*100}{42*1.609} )(\frac{1}{100}) \frac{liter}{km}\\\\= (\frac{3.79*100}{42*1.609} ) \frac{liter}{100\ km}\\\\= 5.61 (\frac{liter}{100\ km})\\\\= \frac{5.61 \ liters}{100 \ km}[/tex]
Also, Nanette's family car rating, in liters per 100 km = 6 liters / 100km.
Thus, Sadie's family car uses lesser fuel, because of this Sadie's family car is more fuel efficient.
I Need help on this please
Answers
Answer:
it would be 10.
Step-by-step explanation:
you have to reflect the quadrilateral to Q over x-axis then the Y-axis, and you would see that 10 is on the same side of ST.
when cutting carrots into small pieces, you are deciding whether to use a knife and cutting board or a food processor. Each carrot chopped by hand takes 45 seconds. each carrot chopped in a food processor takes 12 seconds. it takes 25 seconds to clean off the knife and board. It takes 6 minutes to clean the food processor. What is the minimum number of carrots to cut up for which it will be faster to use a food processor than the knife and cutting board? (factor in both the cutting time and the cleaning time, Your final answer should be an integer
Answers
The minimum number of carrots to cut up for which it will be faster to use a food processor than the knife and cutting board is 11.
In order to find the minimum number of carrots to cut up for which it will be faster to use a food processor than the knife and cutting board, we need to calculate the time taken by each method and then compare it. Let's say the minimum number of carrots is x.
Here's how we can calculate the time taken to chop x carrots:
Time taken using knife and cutting board = 45x + 25 (25 is added for the time taken to clean the knife and board)
Time taken using food processor = 12x + 360 (360 is added for the time taken to clean the food processor)
Now, we need to find the value of x for which the time taken using the food processor is less than the time taken using the knife and cutting board.
So,12x + 360 < 45x + 2512x - 45x < 25 - 360-33x < -335x > 10.15
Therefore, the minimum number of carrots to cut up for which it will be faster to use a food processor than the knife and cutting board is 11.
Thus, the minimum number of carrots to cut up for which it will be faster to use a food processor than the knife and cutting board is 11.
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A colony of bacteria has 10,000 bacteria and is growing exponentially. The inequality 10,000(4x) ≥ 320,000 represents the hours, x, for which the colony will have at least 320,000 bacteria.
Which interval represents the hours when the colony will have at least 320,000 bacteria?
[2.5, ∞)
[3, ∞)
[32, ∞)
[0, 2.5]
Answers
Answer:
A [2.5, ∞)
Step-by-step explanation:
Answer:
A, 1st option
Step-by-step explanation:
Find the surface area of the sphere, where r = 10. Give the exact answer and an approximation to the nearest tenth. The measurements are in inches. See Example 6. in² = in² Find the total surface area of the right trapezoidal prism, where b₁ = 16, b₂ = 11, x = 12, y = 11, and h = 16. The measurements are in feet. ft² b₁ h b₂ Trapezoid
Answers
The total surface area of the right trapezoidal prism is 536 square feet. Therefore, the approximate surface area of sphere is 1,256.6 square inches. The surface area of right trapezoidal prism is 1,485 square feet.
Surface area of the sphere:
The formula for the surface area of a sphere is given by A = 4πr², where r is the radius of the sphere. Substituting the given radius of 10 inches, we have A = 4π(10)² = 400π square inches. The exact value is 400π square inches. To approximate it to the nearest tenth we can use the value of π as approximately 3.14159. Therefore, the approximate surface area is 400π ≈ 400(3.14159) = 1,256.6 square inches.
Surface area of the right trapezoidal prism:
The total surface area of a right trapezoidal prism can be calculated by adding the areas of all its faces. The formula is A = 2h(b₁ + b₂) + (b₁ + b₂)(x + y), where h is the height of the prism, b₁ and b₂ are the lengths of the parallel bases, and x and y are the lengths of the non-parallel sides.
Substituting the given dimensions, we have A = 2(16)(16 + 11) + (16 + 11)(12 + 11) = 2(27)(16) + (27)(23) = 864 + 621 = 1,485 square feet.
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4x + 2y = 18 solve for y
Answers
y=5
4x + 2y
4x= 4 x 2= 8
2y= 2 x 5= 10
10 + 8 = -8
On a grid, what is the
distance between two
points located at
(-4,-5) and (-1,-10)?
Answers
Answer:
-3
Step-by-step explanation:
All you have to do is graph it... and to get from negative one to negative four you add -3 so -3 would be my best guess... if this a multiple choice and -3 is not on there then do -4 as the answer.
What is the value of n in the equation below?
12)
125
124
-20
9
O 20
Answers
Answer:
C. 9
Step-by-step explanation:
When dividing exponents with the same base, always subtract the top from the bottom, which in this case would be 9-5=4
The equation is an exponential equation, and the value of n in the equation is 9
How to determine the value of n?The equation is given as:
[tex]\frac{12^n}{12^5} = 12^4[/tex]
Apply the law of indices, to the equation
[tex]12^{n-5} = 12^4[/tex]
Cancel out the common base in the equation
n - 5 = 4
Add 5 to both sides
n = 9
Hence, the value of n in the equation is 9
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Elena bikes 20 minutes each day for exercise. Write an equation to describe the relationship between her distance in miles, D, and her biking speed, in miles per hour, when she bikes: at a constant speed of 13 miles per hour for the entire 20 minutes
Answers
Answer:
D= 3.9 milesStep-by-step explanation:
Given data
speed v= 13 miles per hour
time taken t= 20min to hour 0.3 hours
the distance D
We know that the relationship between speed, time, and distance
speed=distance/time
13=D/0.3
(the expression above describes the relationship between distance, speed, and time)
making D subject of formula we have
D= 0.3*13
D= 3.9 miles
hence the distance is 3.9miles
Answer:
If your looking for the equation it will be
D = 13 · 1/3
And the answer to this equation is 3.9 miles
Hope this helps
closes integers to 4
Answers
Answer:
3 and 5
Step-by-step explanation:
You have to realize that Integers is another word for numbers
1. Which value of x makes the equation 7x - 4 = -4 true?
Answers
Answer:
0
Step-by-step explanation:
7x - 4 = -4
add 4 to both sides
7x = 0
x = 0
Answer:
x = 0
Step-by-step explanation:
Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.
PEMDAS is the order of operation and stands for:
Parenthesis
Exponents (& Roots)
Multiplication
Division
Addition
Subtraction
First, add 4 to both sides of the equation:
7x - 4 (+4) = -4 (+4)
7x = -4 + 4
7x = 0
Next, divide 7 from both sides of the equation:
(7x)/7 = (0)/7
x = 0/7
x = 0
0 is your answer.
~
Prove whether the quadrilateral is a parallelogram using the Distance & Slope Formulas D(-5, -6), E(5, 2), F(4, -4), G(-6, -12)
Answers
To prove that quadrilateral is a parallelogram using the Distance & Slope Formulas D(-5, -6), E(5, 2), F(4, -4), G(-6, -12), follow these steps
:Step 1: Plot the points in a coordinate plane: (D) (-5,-6), (E) (5,2), (F) (4,-4), (G) (-6,-12)
Step 2: Find the distance of all sides using distance formula.
Distance Formula =sqrt((x2 - x1)² + (y2 - y1)²)DE
= sqrt((5 - (-5))² + (2 - (-6))²)
=sqrt(10²+8²)
=sqrt(164)
= 2 sqrt(41)EF
= sqrt((4 - 5)² + (-4 - 2)²)
=sqrt((-1)²+(-6)²)
=sqrt(37)FG
= sqrt((-6 - 4)² + (-12 - (-4))²)
=sqrt((-10)²+(-8)²)
= 2sqrt(41)DG
= sqrt((-6 - (-5))² + (-12 - (-6))²)
=sqrt((-1)²+(-6)²)
=sqrt(37)
Since the opposite sides are equal, we can say that DE = FG and DG = EF.
Step 3: Find the slope of all sides using slope formula.
Slope Formula = (y2-y1)/(x2-x1)DE
= (2-(-6))/(5-(-5))
=8/10=4/5FG
= (-12-(-4))/(-6-4)=-8/-10=4/5DG = (-6-(-12))/(-5-(-6))=6/1=6EF = (-4-2)/(4-5)=-6/-1=6
Since opposite sides have the same slope, DE // FG and DG // EF. Therefore, we can conclude that the quadrilateral is a parallelogram.In conclusion, the quadrilateral is a parallelogram because opposite sides are parallel.
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which situation can be modeled by equation y=3x25
Answers
What are the steps you take to solve the problem?
-2(x – 3) + 12 = 10
Answers
7. Use the Composite Trapezoidal rule with the indicated values of \( n \) to approximate the following integrals. (1 mark) (a) \( \int_{1}^{2} x \ln x d x, \quad n=4 \) (b) \( \int_{2}^{2} x^{3} e^{x
Answers
The Composite Trapezoidal rule is used to approximate the given integrals. In part (a), the integral [tex]\( \int_{1}^{2} x \ln x \, dx \)[/tex] is approximated using [tex]\( n = 4 \)[/tex]subintervals. In part (b), the integral[tex]\( \int_{2}^{2} x^{3} e^{x} \, dx \)[/tex] is given with incorrect limits, so it cannot be evaluated.
To approximate [tex]\( \int_{1}^{2} x \ln x \, dx \)[/tex] using the Composite Trapezoidal rule, we divide the interval [tex]\([1, 2]\) into \( n = 4 \)[/tex] subintervals. The step size, [tex]\( h \)[/tex], is calculated as[tex]\( h = \frac{b-a}{n} = \frac{2-1}{4} = \frac{1}{4} \)[/tex]. Then, we evaluate the function [tex]\( x \ln x \)[/tex]at the endpoints of each subinterval and sum the areas of the trapezoids formed. The approximation formula for the Composite Trapezoidal rule is: [tex]\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[ f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) \right]\][/tex]
Using this formula, we can calculate the approximation for the given integral. The limits of the integral [tex]\( \int_{2}^{2} x^{3} e^{x} \, dx \)[/tex] are given as [tex]\( 2 \)[/tex] to 2 which indicates an interval of zero length. In this case, the integral cannot be evaluated since there is no interval over which to integrate.
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Six less than seven times a number is negative is twenty
Answers
Answer:
7n - 6 = -20
Step-by-step explanation:
how do you calculate monthly forecasting 3 month moving
average
Answers
To calculate a three-month moving average for monthly forecasting, you need to follow these steps: Gather the historical data, Determine the time period, Calculate the moving average, Repeat the process.
Gather the historical data: Collect the monthly data for the specific variable you want to forecast. For example, if you want to forecast sales, gather the sales data for the past several months.
Determine the time period: Decide on the time period for your moving average. In this case, it is three months.
Calculate the moving average: Add up the values for the variable you are analyzing over the past three months and divide the sum by three to get the average. This will be your moving average value for the third month.
Repeat the process: Shift the time period by one month and calculate the moving average for the new three-month period. Continue this process for each subsequent month, updating the time period and calculating the moving average accordingly.
For example, let's say you have the following sales data for the past six months:
Month 1: 100 units
Month 2: 120 units
Month 3: 110 units
Month 4: 130 units
Month 5: 140 units
Month 6: 150 units
To calculate the three-month moving average for Month 4, you would add up the sales values for Month 2, Month 3, and Month 4 (120 + 110 + 130 = 360) and divide it by three to get an average of 120 units. Repeat this process for each subsequent month to obtain the moving average values for your forecast.
Note that the number of data points you include in the moving average calculation and the frequency of the data (monthly in this case) can be adjusted based on your specific needs and the nature of the data.
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f(x) = -3x2+1
If the graph of fis translated vertically upward by 5 units, it becomes the graph of a function h.
Find the expression for h (x).
Answers
What is the sum of these fractions?
3/8 + 1/6
A. 5/25
B. 4/14
C. 13/24
D. 18/14
Answers
Answer: C. 13/24 i hope it help you out
What is a x-coordinate
Answers
Answer:
The horizontal line on a graph.
Step-by-step explanation:
Answer:
The horizontal value in a pair of coordinates: how far along the point is.
The X Coordinate is always written first in an ordered pair of coordinates (x,y), such as (12,5).
I hope this help you out
10. Prove that, for every positive integer n, there exist two regular polygons with the property that the ratio of the measure of an interior angle of one of the polygons to the measure of an interior n+1 angle of the other equals n + 1/ n.
Answers
It's given to prove that, for every positive integer n, there exist two regular polygons with the property that the ratio of the measure of an interior angle of one of the polygons to the measure of an interior n + 1 angle of the other equals n + 1/ n.
We know that for a regular polygon with n sides, the measure of each interior angle is given by:
(n-2)×180°/n
That is to say, each interior angle of a regular polygon with n sides is equal to (n-2)×180/n degrees.
Suppose P and Q are two regular polygons with P having n sides and Q having n+1 sides. Then, measure of each interior angle of polygon P is given by: (n-2)×180/n degrees
Similarly, the measure of each interior angle of polygon Q is given by: (n-1)×180/(n+1) degrees
Thus, the ratio of the measure of an interior angle of one of the polygons (say P) to the measure of an interior n+1 angle of the other (say Q) is given by:
(n-2)×180/n ÷ (n-1)×180/(n+1) = (n+1)/n
Therefore, we can conclude that for every positive integer n, there exist two regular polygons with the property that the ratio of the measure of an interior angle of one of the polygons to the measure of an interior n+1 angle of the other equals n+1/n.
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[ 2 8] [2 0] [7 2]
Given A = [3 0] B = [3 8] and C = [6 3] [5 1] (a) Is AB def ned? Calculate AB. Can you calculate BA ? Why?
(b) Is BC defined? Calculate BC. Is CB defined? If so, calculate CB. Is it true that BC=CB ?
Answers
The exercise involves calculations with matrices A, B, and C. We need to determine if matrix products AB, BA, BC, and CB are defined. Additionally, we need to calculate the specific matrix products and determine if BC is equal to CB.
Let's perform the calculations step by step:
(a) To determine if AB is defined, we need to check if the number of columns in matrix A is equal to the number of rows in matrix B. If they are equal, the matrix product AB is defined. If not, the product is not defined.
Matrix A has 2 columns and matrix B has 2 rows, so the product AB is defined.
Calculating AB:
AB = [23 + 83 20 + 88]
[23 + 03 20 + 08]
= [30 64]
[6 0]
To calculate BA, we need to check if the number of columns in matrix B is equal to the number of rows in matrix A. In this case, matrix B has 2 columns and matrix A has 2 rows, so the product BA is also defined.
Calculating BA:
BA = [32 + 02 38 + 02]
[32 + 02 38 + 02]
= [6 24]
[6 24]
(b) To determine if BC is defined, we need to check if the number of columns in matrix B is equal to the number of rows in matrix C. In this case, both matrices have 2 columns and 2 rows, so the product BC is defined.
Calculating BC:
BC = [26 + 85 23 + 81]
[06 + 05 03 + 01]
= [52 22]
[0 0]
To calculate CB, we need to check if the number of columns in matrix C is equal to the number of rows in matrix B. In this case, matrix C has 2 columns and matrix B has 2 rows, so the product CB is also defined.
Calculating CB:
CB = [62 + 30 68 + 30]
[52 + 10 58 + 10]
= [12 48]
[10 40]
Is it true that BC = CB?
BC = [52 22]
[0 0]
CB = [12 48]
[10 40]
BC and CB are not equal. The matrix product is not commutative, so BC and CB may yield different results.
In summary:
(a) AB is defined and the result is:
AB = [30 64]
[6 0]
BA is also defined and the result is:
BA = [6 24]
[6 24]
(b) BC is defined and the result is:
BC = [52 22]
[0 0]
CB is also defined and the result is:
CB = [12 48]
[10 40]
BC and CB are not equal, indicating that matrix multiplication is not commutative.
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