How to find left and right inverse of a 3x2 matrix.

Answers

Answer 1

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]


Related Questions

PLEASE ANSWER THIS RIGHT ASAP

Answers

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 :)

23= 7/9(6x-36)+9
<=>23= 14/3x-83/3+9
<=>43= 14/3x
<=> x= 9
Ok done. Thank to me :>

Vanessa 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)

Answers

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

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

Answers

Answer:

0,2%

Step-by-step explanation:

16 / 8000 = 0,002

0,002 * 100 % = 0,2%

1\4-2\7 plese show your work

Answers

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.

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?​​

Answers

Answer:

Write about like something that happened at school that involved you and others.

Step-by-step explanation:

Hope this helps!

You can write about a story that happened with others around you?? Or an story that happened to you once??

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.

Answers

C. They will have the same solution because they represent the same lines when plotted on the coordinate axes

(Pythagorean theorem) do the following lengths form a right triangle? (Show steps)

Answers

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

Juana buys a computer for $785.00. Her local sales tax is 6.85%. How much does Juana pay for the computer, including tax?

Answers

Answer:

838.7725

Step-by-step explanation:

Please help me with this homework …

Answers

Answer:

I think letter B

Step-by-step explanation:

Yyyyyyyygggggggggggggggg

solve pls brainliest

Answers

1 no
2 no
3 no
4 no
Hope it helps! :)

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.

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?

Answers

Answer:

$125 per month

Step-by-step explanation:

(Pythagorean theorem) find the missing side length. Round to the nearest hundredth (show steps)

Answers

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

What is the solution for the inequality 4≤r/18+10?

Answers

[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]

what is the answer ?????​

Answers

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]

Find the slope of the line graphed below.​ 21 points!

Answers

The answer is 3/2
3 units up and 2 units to the left that is the slope

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!

helpppppppppppppppppppppppppppppppp

Answers

Answer:

x ≤ -14

Step-by-step explanation:

plz help me fast Simplify t12/t6

Answers

Answer:

t^6

Step-by-step explanation:

when dividing exponents, you subtract (refer to exponent rule)

t^12 - t^6 = t^6

How many decimals in multiplication.

Answers

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.

Simplify the expression: 5+2a - (9+a)

Answers

Answer:

a - 4

Step-by-step explanation:

[tex]5+2a-(9+a)\\5+2a-9-a\\a-4[/tex]

€1 = £0.85. Change €345 into £.

Answers

Answer:

£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.

Solve the given equation

x – 1 + 5x = 23

Answers

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.

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.

Answers

Answer:

See below

Step-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 - hypotenuse

45×45×90 right triangle has side ratios as:

l : l : h = 1 : 1 : √2, where l- legs, h- hypotenuse

If 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√3

b) 45×45×90 right triangle

Given l = √22BC = l = √22AB = l√2 = √22×√2= √44 = 2√11

c)  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√2

0.125 ultiply by 1000

Answers

Answer:

125 is answer .........:)

help will give brainliest

Answers

Answer: the answer is line 2

Step-by-step explanation:

12x+3=6 4x+1=2
what is happening in this step of the equation solving process?

Answers

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

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).

Answers

Answer:

4.7

Step-by-step explanation:


Select the values that make the inequality h < 1 true.
(Numbers written in order from least to greatest going across.)

Answers

H = 0 and all negative numbers

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.

Answers

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!!

Will give brainliest if right

Answers

Answer:

the answer is d to the problem

2 iPhones cost $1,680, how much would 10 iPhones cost?

Answers

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

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