Let A be a 3×2 matrix, L its left inverse, and R its right inverse. L and R are then matrices such that LA = I₂ (the 2×2 identity matrix) and AR = I₃ (the 3×3 identity matrix). Clearly L must be 2×3 and R must be 3×2 in order for the matrix products to be defined.
To find L and R, we start by introducing a square matrix on the the left sides of either equation above. In particular, we uniformly multiply both sides by the transpose of A, then solve for the inverse.
For the left inverse, we have
[tex]LA=I[/tex]
[tex](LA)A^\top = IA^\top[/tex]
[tex]L\left(AA^\top\right) = A^\top[/tex]
[tex]\left(L\left(AA^\top\right)\right)\left(AA^\top\right)^{-1} = A^\top \left(AA^\top\right)^{-1}[/tex]
[tex]L\left(\left(AA^\top\right)\left(AA^\top\right)^{-1}\right) = A^\top \left(AA^\top\right)^{-1}[/tex]
[tex]LI = A^\top \left(AA^\top\right)^{-1}[/tex]
[tex]L = A^\top \left(AA^\top\right)^{-1}[/tex]
We do the same thing for the right inverse, but take care with how we multiply both sides of AR = I₃.
[tex]AR=I[/tex]
[tex]A^\top(AR)=A^\top I[/tex]
[tex]\left(A^\top A\right)R = A^\top[/tex]
[tex]\left(A^\top A\right)^{-1} \left(\left(A^\top A\right)R\right) = \left(A^\top A\right)^{-1} A^\top[/tex]
[tex]\left(\left(A^\top A\right)^{-1} \left(A^\top A\right)\right) R = \left(A^\top A\right)^{-1} A^\top[/tex]
[tex]IR = \left(A^\top A\right)^{-1} A^\top[/tex]
[tex]R = \left(A^\top A\right)^{-1} A^\top[/tex]
There is a bag filled with 5 blue and 6 red marbles.
A marble is taken at random from the bag, the colour is noted and then it is replaced.
Another marble is taken at random.
What is the probability of getting at least 1 blue?
Answer:
5 by 11
6 plus 5 is 11
qn says at least 1 should be blue which will by 5 by 11
Answer:
5/11
Step-by-step explanation:
Because theres 5 blue marbles out of 11
True or False: The U.S. Bureau of the Census provides information that is useful for defining the denominator in rates.
Answer:
True
Step-by-step explanation:
I really need help right now please i will mark brainliest and give lots of points
Answer:
What do you need help with?
Step-by-step explanation:
Answer:
what is the problem?
Step-by-step explanation:
Reasoning A store sells packages of comic books with a poster. A poster and 8 comics cost $14.75. A poster and 16 comics cost $22.75. Write a linear function rule that models the cost y of a package containing any number x of comic books. Suppose another store sells a similar package, modeled by a linear function rule with initial value $5.50. Use pencil and paper. Explain which store has the better deal.
The linear function rule is y=
If x = 2, calculate the value of: 2x2 - x
Which inequality is a true statement?
Select each correct answer.
−4 = −5
−4 ≤ −5
−4 < −5
−4 > −5
−4 ≥ −5
Answer:
i think -4>-5
Step-by-step explanation:
because (-4)is greater than (-5)
HELP PLEASE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
Omsimmmmm Nice G UwU
Sarry Sarry
Help me please!!!!!!!!!
Answer:
X=5
Step By Step:
PLS HELP WILL MARK BRAINLIEST!
Answer:
According to me answer is option E
Step-by-step explanation:
If you like my answer than please mark me brainliest thanks
A local business donated 40,782 marbles to distribute equally
among 42 sixth grade classes in the district. How many marbles
did each class receive?
Answer:
971
Step-by-step explanation:
40782/42=971
Answer: 971 marbles per class
Step-by-step explanation:
40,782 marbles divided by 42 classes =
971 marbles per class
Check
971 x 42= 40782
witch rational number is between -11/15 and -5/6?
answers to chose from -1, -7/10, -4/5 and -13/15
Answer:
-4/5
Step-by-step explanation:
To find this we can look for the [tex]LCD[/tex] between all the numbers
Here are our [tex]LCD's[/tex] (Least Common Denominator)
-22/30 and -25/30
It wouldn't be -1 because it isn't in between those numbers
It isn't -7/10 because if we turn it into our [tex]LCD[/tex] it would be -21/30 which isn't in between those numbers
It is -4/5 because if we turn in into our [tex]LCD[/tex] it would be -24/30 which is between those numbers
This means we have our right answer we know our last one is incorrect
Is 70.61 a rational or irrational number?
Answer:
70.61 is a rational number
Step-by-step explanation:
This is because a rational number is either:
A whole number, like 70, whole number.
Is a terminating decimal, such as the one above 70.61, (A terminating decimal is a decimal that doesn't repeat.
And or has a pattern, or a recurring decimal, like 7.478478..., it repeats the decimal over and over, thus, rational.
Hope this helped you, have a great day.
Simplify 7-2 + 2(4 divide 1/2)
Answer:
21
Step-by-step explanation:
5 + 16 = 21
Answer:
21
Step-by-step explanation:
hope it helps
Select the system that has the same solution as
–2x+2y = -5,
5x — бу = 8
The times for running a 5K for a local charity event are Normally distributed with a mean time of 28 minutes and
standard deviation of 5.4 minutes. Which time represents the 60th percentile?
Find the z-table here.
A. 19.6 minutes
B. 26.6 minutes
C. 29.4 minutes
D. 36.4 minutes
Answer:
29.4 minutes represents the 60th percentile
Step-by-step explanation:
Given : The times for running a 5K for a local charity event are Normally distributed, with a mean time of 28 minutes and standard deviation of 5.4 minutes.
To Find : Which time represents the 60th percentile?
19.6 minutes
26.6 minutes
29.4 minutes
36.4 minutes
Solution:
time represents the 60th percentile
Means 60 % data = 0.6 below this
from z table , z score for this = 0.253
z score = ( Value - Mean ) / SD
=> 0.253 = ( Value - 28 ) /5.4
=> 1.3662 = Value - 28
=> Value = 29.3662
=> Value = 29.4
29.4 minutes represents the 60th percentile
A board is 80 inches long. How long is it in feet and inches? please help ;-;
Answer:
6 feet and 8 inches
Step-by-step explanation:
1 foot = 12 inches
Answer: 6ft & 8 inches
Step-by-step explanation: 12 inches = 1 feet
2 feet = 24
3 feet = 36
4 feet = 36
5 feet = 60
6 feet = 72
80 - 72 = 8
6ft & 8 inches
Russel wants to buy 3 identical gift cards. He wants to make sure his total cost is no more than $2 above or below $60. Which of the following inequalities represents this situation?
A|3c − 60| ≤ 2
B|60 + 3c| ≤ 2
C |3c − 2| ≤ 60
D |3c + 2| ≤ 60
Step-by-step explanation:
very imprecise phrasing.
I assume this means the cost should not be more than $2 above, and not less than $2 below $60.
the bottom line is : the difference between the price of the 3 gift cards and the planned budget of $60 must not be more than $2 (no matter what's the sign of the difference - hence the usage of absolute values).
the only answer calculating that difference and comparing it to the $2 threshold is A.
remember, a difference is always calculated by a "-" operation.
so, B and D disqualify.
C compares the price of the 3 cards with $2 and compares that result with the limit of $60.
yes, that protects against too expensive cards.
but no, it does NOT protect against to cheap cards. e.g. if the cards (3c) cost only $30, that inequality would still be true (but should not).
so, A is the correct answer.
if u have 105 in 7 days what would you have in 1 day?
Answer:
I would have 15.
Step-by-step explanation:
In 7 days, I would have 105.
In 1day, I would have 105/7
=15
find slope of line with the points (-5,-4) and has a slope of -3/5
Answer:
-3/5
Step-by-step explanation:
The answer is already in your question. You're asking for the slope of a line with an already stated slope
I If I multiply the sum of 630 and 1453 by zero what will be my result
Answer:
The result will be 0 (zero)
Step-by-step explanation:
The result will be zero because the product of any number and zero is always zero.
(630+1453)×0
= 2083×0
= 0
According to the Empirical Rule, what percentage of a normal population falls between 1 standard deviations left of the mean and 3 standard deviations right of the mean
Answer:
68% of the data falls within one standard deviation, 95% percent within two standard deviations, and 99.7% within three standard deviations from the mean.
Step-by-step explanation:
2. Order the numbers from least to greatest -0.5, -.025, 0.05,-2,5
Answer:
-2, -0.5, -.025, 0.05, 5
Identify the slope (m) and y-intercept (b) for 3x + 4y = 9.
Answer:
slope= -¾
[tex]y - intercept = 2 \frac{1}{4} [/tex]
Step-by-step explanation:
3x +4y= 9
Let's rewrite the equation into the form of y= mx +b (known as slope-intercept form), where m is the slope and b is the y-intercept.
Keep the y term on the left-hand side of the equation and move all the other terms to the right-hand side.
4y= -3x +9
For the coefficient of y to be 1, divide both sides of the equation by 4.
[tex]y = - \frac{3}{4} x + \frac{9}{4} [/tex]
[tex]y = - \frac{3}{4} x + 2 \frac{1}{4} [/tex]
[tex]slope = - \frac{3}{4} [/tex]
[tex]y - intercept = 2 \frac{1}{4} [/tex]
The graph represents the distance a car traveled over time.
What is the constant of proportionality K.
Answer:
K=2/5
Step-by-step explanation:
So, I’m assuming that by constant of proportionality you mean slope of the line. To do that, you just figure out how y changes with x. The easiest way to do this is rise over run (y over x). If you look at the graph, there is a point that pass through (0,0) and another that passes through (5,2) so if you count how many spaces we moved up then how many spaces we moved right, you will get the slope of the line. Spaces moved up are 2-0=2 and spaced moved right are 5-0=5. Use rise over run which is going to be 2/5. So, K=2/5.
solve the system of equations by elimination:
3x+y=5
2x-2y=-2
Answer:
(x, y) = (1, 2)
Step-by-step explanation:
The method of elimination works best when the coefficient of a variable in one equation is equal or opposite the coefficient of the same variable in the other equation.
Here, we notice that all of the coefficients of the second equation are multiples of 2. When we divide that equation by 2, we get ...
x -y = -1
This equation has a y-coefficient that is the opposite of the y-coefficient in the first equation. We can eliminate the y-variable by adding these equations together.
(3x +y) +(x -y) = (5) +(-1)
4x = 4 . . . . . . . simplify. Notice the y-variable has been eliminated.
x = 1 . . . . . . . . divide by 4
Using this value of x in the second equation, it tells us ...
1 -y = -1
Adding y+1 to both sides of this equation gives ...
2 = y
The solution to the system of equations is (x, y) = (1, 2).
__
The graphing calculator confirms this solution.
25) 2x + 3y = 9
find slope and y intercept
Answer: Slope = -2/3 Y-intercept = (0,3)
Step-by-step explanation: Use the slope-intercept form y = mx + b to find the slope m and y-intercept b.
Decompose and use the distributive property to find the product of 5 × 7.3. 35.0 36.5 35.5 31.5 PLEASE HELP PLEASE T^T
Answer:
The answer is 36.5
Step-by-step explanation:
8x - 2y = 4 and x + 4y = -12
x = -12 - 4y
y = -3 - (x/4)
There you go! :D
This table gives select values of the differentiable function h.
x -4 -1 0 1 4
h(x) -26 -15 -32 -39 -35
What is the best estimate for h'(-2) we can make based on this table?
Choose 1 answer:
a 11
b 3.67
c -1.13
d -20.5
11 times 11 in binary . Help i need it today!
Answer:
1011×1011 = 11110011
Step-by-step explanation:
11 in binary is 1011
121 in binary is 1111001