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]
What is the solution for the inequality 4≤r/18+10?
[tex]4 \leq \dfrac{r}{18} +10\\\\\\\implies 4-10 \leq \dfrac{r}{18}\\\\\\\implies -6 \leq \dfrac r{18}\\\\\\\implies -6(18) \leq r\\\\\\\implies -108 \leq r\\\\\\\implies r \geq -108\\\\\\\text{Interval,}~ [-108 , \infty)[/tex]
1\4-2\7 plese show your work
Answer:
-1/28
Step-by-step explanation:
You need to convert 2/7 and 1/4 into the same denominator. The GCD is 28. 2/7=8/28 and 1/4=7/28. 7/28-8/28=-1/28
Answer:
-1/28
Step-by-step explanation:
You have to find the least common multiple for 7 and 4, the denominators, to add the fractions. The LCM of 7 and 4 would be 28. You would have to multiply 7 by 4 and 4 by 7, meaning you have to multiply the numerators by the same number. 1x7/4x7-2x4/7x4= -1/28.
2 iPhones cost $1,680, how much would 10 iPhones cost?
Answer:
$8,400
Step-by-step explanation:
one way to solve it would be that since you know 2 x 5 = 10 you can do $1,680 x 5 and that = $8,400
Find the slope of the line graphed below. 21 points!
Answer:
The answer is 1.5
Step-by-step explanation
First you want to find the rise. This is the amount of height between the two points (Y) and divide them over the run. This is amount of distance (X) between the points. This will be 6/4. Divide this and you have your answer!
Simplify the expression: 5+2a - (9+a)
Answer:
a - 4
Step-by-step explanation:
[tex]5+2a-(9+a)\\5+2a-9-a\\a-4[/tex]
(Pythagorean theorem) do the following lengths form a right triangle? (Show steps)
Answer:
yes
Step-by-step explanation:
Pythagorean theorem equation is a²+b²=c²
c is the longest side which is 5. a and b dont matter in this case
3²+4²=5²
9+ 16 = 25..........simply
25 = 25...............true, these numbers form a right triangle
12x+3=6 4x+1=2
what is happening in this step of the equation solving process?
Step-by-step explanation:
12x=6-3
12x=3
12x/12=3/12
x=1/4
4x+1=2
4x=2-1
4x=1
4x/4=14
x=1/4
I you have great day
help will give brainliest
Answer: the answer is line 2
Step-by-step explanation:
helpppppppppppppppppppppppppppppppp
Answer:
x ≤ -14
Step-by-step explanation:
(Pythagorean theorem) find the missing side length. Round to the nearest hundredth (show steps)
Answer:
The missing length is 10.44
Step-by-step explanation:
a^2 + b^2 = c^2
3^2 + 10^2 = c^2
9 + 100 = c^2
109 = c^2
√c^2 = √109
c = √109
c ≈ 10.440307
c ≈ 10.44
Answer:
10.44
Step-by-step explanation:
Pythagorean theorem equation is a²+b²=c²
3²+10²=c²
9+100=c²........simplify
109=c².............add
√109=c
10.44031
10.44................round to the nearest hundredth
PLEASE TELL ME A STORY I WANT AN IDEA ON WHAT TO WRITE NOT JUST A SIMPLE ANSWER LIKE I GAVE A HOMELESS MAN MONEY Describe a time when you solved a problem. What actions did you take, and what impact did you make?
Answer:
Write about like something that happened at school that involved you and others.
Step-by-step explanation:
Hope this helps!
help asap for brainlist
Given g(x)=−6x+3, find g(−4). To earn full credit, show your work using a method explained in the Orientation.
Answer:
g(-4) = 24 + 3 = 27
Step-by-step explanation:
We find the value of g(-4) by substituting -4 for x in g(x)= −6x + 3:
g(-4) = -6(-4) + 3, or
g(-4) = 24 + 3 = 27
Answer:
27
Step-by-step explanation:
substitute -4 for x
-6(-4) + 3
24 + 3
27
hope it helps :)
mark brainliest!!
Solve the given equation
x – 1 + 5x = 23
Step-by-step explanation:
x - 1 + 5x = 23
6x - 1 = 23
6x = 23 + 1 = 24
6x = 24
x = 24/6
x = 4 ans.
HOPE THIS ANSWER HELPS YOU DEAR! TAKE CARE.
solve pls brainliest
Answer:
0.92 can be written as a fraction, because it is a terminating decimal. ( See the following. )
[tex] \frac{23}{25} [/tex]
0.777... can't be wrtten as a fraction, because it is a repeating decimal.
0.444... can't be written as a fraction, because it is a repeating fraction.
0.5813.. can't be written as a fraction, because
it is a decimal that neither terminates nor repeats.
For example : Root numbers don't terminates or repeats.
[tex] \sqrt{ 2} = 1.41421...[/tex]
I hope it helps.
How many decimals in multiplication.
Answer:
In multiplication of a decimal by a decimal number we will learn how to find the product of a decimal by another decimal.
To multiply a decimal number by a decimal number, we first multiply the two numbers ignoring the decimal points and then place the decimal point in the product in such a way that decimal places in the product is equal to the sum of the decimal places in the given numbers.
PLEASE HELP!!!!! I NEED THE ANSWER SOON
Two systems of equations are shown:
System A System B
3x + 2y = 3 −x − 14y = 1
−2x − 8y = −1 −2x − 8y = −1
Which of the following statements is correct about the two systems of equations?
A.)The value of x for System B will be one-third of the value of x for System A because the coefficient of x in the first equation of System B is one third times the coefficient of x in the first equation of System A.
B.) The value of x for System A will be equal to the value of y for System B because the first equation of System B is obtained by adding −4 to the first equation of System A and the second equations are identical.
C.) They will have the same solution because they represent the same lines when plotted on the coordinate axes.
D.)They will have the same solution because the first equation of System B is obtained by adding the first equation of System A to 2 times the second equation of System A.
A company that makes hair-care products had 8,000 people try a new shampoo. Of the 8,000 people, 16 had a mild allergic reaction. What percent of the people had a mild allergic reaction
Answer:
0,2%
Step-by-step explanation:
16 / 8000 = 0,002
0,002 * 100 % = 0,2%
plz help me fast Simplify t12/t6
Answer:
t^6
Step-by-step explanation:
when dividing exponents, you subtract (refer to exponent rule)
t^12 - t^6 = t^6
Juana buys a computer for $785.00. Her local sales tax is 6.85%. How much does Juana pay for the computer, including tax?
Answer:
838.7725
Step-by-step explanation:
Give the simplified radical form for each side length requested.
Pls answer only if you can explain and if your sure it’s the right answer.
I need help in understanding how to work it myself so please make sure you explain well.
Answer:
See belowStep-by-step explanation:
30×60×90 right triangle has side ratios as:
s : l : h = 1 : √3 : 2, where s- short leg, l - long leg, h - hypotenuse45×45×90 right triangle has side ratios as:
l : l : h = 1 : 1 : √2, where l- legs, h- hypotenuseIf you know one of the sides its easy to find the missing ones.
a) 30×60×90 right triangle
Given l = 12s : 12 : h = 1 : √3 : 2CB = s = 12 / √3 = 12√3/3 = 4√3 AB = h = 2s = 2*4√3 = 8√3b) 45×45×90 right triangle
Given l = √22BC = l = √22AB = l√2 = √22×√2= √44 = 2√11c) 30×60×90 right triangle
Given h = 20√6s : l : 20√6 = 1 : √3 : 2BC = s = h/2 = 20√6/2 = 10√6AC = l = s√3 = 10√6×√3 = 10√18 = 10×3√2 = 30√2Vanessa walks her puppy 12 of a miles in 20 minutes. How far does she walk her puppy in 1 hour?
Show work. (Type answer in word form)
Answer:
thirty-six of a miles or [tex]36[/tex] miles of a miles <--- I didn't know if you meant regular miles
Step-by-step explanation:
We can convert 1 hour to 60 minutes
[tex]60\div20=3[/tex]
Since it is [tex]3[/tex] [tex]times[/tex] the time and we assume her rate is consistent we multiply 3 by 12
[tex]3\cdot12=36[/tex] of a miles
thirty-six of a miles
Brainliest please :)
Answer:
Vanessa walks her puppy for 36 miles in 1 hour.
Step-by-step explanation:
She walks 12 miles in 20 minutes
1 hour = 60 minutes
There are 3 - 20 minute periods so she walks her puppy 12(3) miles, or 36 miles
-Chetan K
Select the values that make the inequality h < 1 true.
(Numbers written in order from least to greatest going across.)
what is the answer ?????
Q. 1 [tex] \frac{19}{20} \times 100 = 19 \times 5 = 95\% \\ [/tex]
Q. 2). 30 - 24 = 6
[tex] \frac{6}{30} \times 100 = \frac{1}{5} \times 100 = 20 \\ [/tex]
Consider the line y=3x-3 .
Find the equation of the line that is parallel to this line and passes through the point(7, 4) .
Find the equation of the line that is perpendicular to this line and passes through the point (7, 4).
Answer:
4.7
Step-by-step explanation:
Will give brainliest if right
Answer:
the answer is d to the problem
0.125 ultiply by 1000
Answer:
125 is answer .........:)
€1 = £0.85. Change €345 into £.
£293.25
Step-by-step explanation:If 1 euro is worth 0.85 pounds, the total of 345 euros is 0.85 times the value of a single euro. This is therefore 0.85 * 345, or £293.25.
Please help me with this homework …
Answer:
I think letter B
Step-by-step explanation:
Yyyyyyyygggggggggggggggg
PLEASE ANSWER THIS RIGHT ASAP
Answer:
x=9
Step-by-step explanation:
23=7/9(6x-36)+9
Distribute 7/9.
23=14/3x-28+9
Add.
23=14/3x-19
Add 19 to both sides of the equation.
42=14/3x
Divide 14/3 from both sides of the equation.
X=9
Hope this helps :)
Ryan made a $3000 down payment on a car. The total cost of the car was $7500. He made 36 equal monthly payments to pay the car in full. How much did Ryan pay per month?
Answer:
$125 per month
Step-by-step explanation:
