If these vectors are to be linearly independent, then there exists some not-all-zero choice for scalars [tex]c_1,c_2,c_3,c_4[/tex] such that
[tex]c_1(1,1,0,-1) + c_2(1,k,1,1) + c_3(5,1,k,1) + c_4(-1,1,1,k) = (0,0,0,0)[/tex]
We can recast this as a system of linear equations,
[tex]\begin{bmatrix}1&1&5&-1\\1&k&1&1\\0&1&k&1\\-1&1&1&k\end{bmatrix}\begin{bmatrix}c_1\\c_2\\c_3\\c_4\end{bmatrix} = \begin{bmatrix}0\\0\\0\\0\end{bmatrix}[/tex]
This system has a solution if the coefficient matrix on the left side is not singular. So we attempt to find k such that it *is* singular, i.e. the determinant is zero:
• A cofactor expansion along the first column gives
[tex]\begin{vmatrix}1&1&5&-1\\1&k&1&1\\0&1&k&1\\-1&1&1&k\end{vmatrix} = \begin{vmatrix}k&1&1\\1&k&1\\1&1&k\end{vmatrix} - \begin{vmatrix}1&5&-1\\1&k&1\\1&1&k\end{vmatrix} + \begin{vmatrix}1&5&-1\\k&1&1\\1&k&1\end{vmatrix}[/tex]
• Cofactor expansions along the first columns of the remaining 3x3 matrices give
[tex]\begin{vmatrix}k&1&1\\1&k&1\\1&1&k\end{vmatrix} = k \begin{vmatrix}k&1\\1&k\end{vmatrix} - \begin{vmatrix}1&1\\1&k\end{vmatrix} + \begin{vmatrix}1&1\\k&1\end{vmatrix} = k^3-3k+2[/tex]
[tex]\begin{vmatrix}1&5&-1\\1&k&1\\1&1&k\end{vmatrix} = \begin{vmatrix}k&1\\1&k\end{vmatrix} - \begin{vmatrix}5&-1\\1&k\end{vmatrix} + \begin{vmatrix}5&-1\\k&1\end{vmatrix} = k^2-4k+3[/tex]
[tex]\begin{vmatrix}1&5&-1\\k&1&1\\1&k&1\end{vmatrix} = \begin{vmatrix}1&1\\k&1\end{vmatrix} - \begin{vmatrix}5&-1\\k&1\end{vmatrix} + \begin{vmatrix}5&-1\\1&1\end{vmatrix} = -k^2 - 6k + 7[/tex]
It follows that
[tex]\begin{vmatrix}1&1&5&-1\\1&k&1&1\\0&1&k&1\\-1&1&1&k\end{vmatrix} = (k^3-3k+2) - (k^2-4k+3) + (-k^2 - 6k + 7) \\\\ \begin{vmatrix}\cdots\end{vmatrix} = k^3 - 2k^2 - 5k + 6 = (k-3)(k-1)(k+2)[/tex]
which makes the coefficient matrix singular if k = 3, k = 1, or k = -2.
Then the four vectors are linearly independent for
[tex]\left\{ k\in\mathbb R \mid k\not\in\{-2,1,3\}\right\}[/tex]
A store advertises a sale on t-shirts. The advertised price is four t-shirts for $48. Write and solve an equation to find the cost of one t-shirt.
a.) $8
b.) $12
c.) $16
d.) $48
Help mee plsssssssdds
Answer:
No Inez is not correct.
Step-by-step explanation:
[tex]\frac{1}{3} > \frac{1}{5}[/tex] (1/3 is greater than 1/5)
Help thank you………………
Answer:
it is too easy I will answer it soon.
If 18 drinks cost £54, how much will 7 drinks cost ?
Answer:
£21
Step-by-step explanation:
Cost of 18 drinks = £54
Cost of 1 drink
= Cost of 18 drinks/18
= £54/18
= £3
Cost of 7 drinks
= Cost of 1 drink × 7
= £3 × 7
= £21
Write an equivalent expression in standard form for
(5x + 4)(7x+ -12x + 27).
Answer: -25x² + 115x + 108
Concept:
When encountering questions that ask for simplifying polynomial expressions, using the method FOIL would be easier:
First, which means multiplying the first terms together;Outer, which means that we multiply the outermost terms when the binomials are placed side by side;Inner, which means multiply the innermost terms together;Last, which means multiplying the last term in each binomial together;Solve:
Given
(5x + 4) (7x + (-12x) + 27)
Combine like terms in the parentheses
=(5x + 4) (7x - 12x + 27)
=(5x + 4) (-5x + 27)
Expand parentheses and apply the FOIL method
=-25x² + 135x - 20x + 108
Combine like terms
= [tex]\boxed {-25x^2+115x+108}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Answer:
[tex]\boxed {\boxed {\sf -25x^2+115x+108}}[/tex]
Step-by-step explanation:
We are asked to find an equivalent expression for (5x + 4)(7x+ -12x + 27). Note that we are multiplying 2 polynomials.
[tex](5x + 4)(7x+ -12x + 27)[/tex]
Notice that the second polynomial has like terms. There are 2 terms with an x (7x and -12x) and we can combine them.
[tex](5x + 4)[(7x+ -12x) + 27][/tex]
[tex](5x + 4)[(-5x) + 27][/tex]
[tex](5x+4)(-5x+27)[/tex]
Now we are multiplying 2 binomials. Therefore, we use the FOIL method. We multiply the First terms, the Outside terms, the Inside terms, and the Last terms.
First: 5x * -5x = -25x²Outside: 5x * 27 = 135x Inside: 4* -5x = -20x Last: 4 * 27=108[tex]-25x^2+135x-20x+108[/tex]
Notice there are 2 terms with an x ( 135x and -20x). They are like terms and can be combined.
[tex]-25x^2+(135x-20x)+108[/tex]
[tex]-25x^2+115x+108[/tex]
The equivalent expression in standard form is -25x²+115x+108
What is the correct clases for this drugs
Compazine
heelp me please
Answer:
It is a addiction XD
Step-by-step explanation:
Bsgsgwgwggwgwfeffefecrcr
pleaseeeeee solve this!
Answer:
C
Step-by-step explanation:
One graph line is equal to 10km and there is 2 graph lines between Red Bank and Belleville.
Which set of ordered pairs represents a function?
{(3,6), (3, 1), (9,0), (2, 2)}
{(2,-5), (-3, 6), (-1, -8), (-3,-4)}
{(4, -7), (-2, -1), (-7,4), (9, -1)}
{(3,8), (-2,-9), (1,4), (1,9)}
Answer:
{(4, -7), (-2, -1), (-7,4), (9, -1)}
Step-by-step explanation:
Every other set of ordered pairs has an x-value that has more than one y-value. This is against the rules of a function. Therefore, {(4, -7), (-2, -1), (-7,4), (9, -1)} represents a function.
Convert this rational number
to its decimal form and round
to the nearest thousandth.
1/6
Answer: 0.166
Step-by-step explanation:
Converting to a decimal:
1/6 = 0.1666 (1 divided by 6 - long division)
Rounding:
0.1666 to the nearest thousandth = 0.166
100 10 times as much as ___ of
Answer:
1 thousand
Step-by-step explanation:
1 ten = 10 x 1 one (Read, as “1 ten is 10 times as much as 1 one.”) 1 hundred = 10 x 1 ten (Say, “1 hundred is 10 times as much as 1 ten.”) 1 thousand = 10 x 1 hundred (Say, “1 thousand is 10 times as much as 1 hundred.”)i hope im right
54394 plus 13768
Thank you
Answer:
[tex]54394 + 13768 \\ \\ = 68162[/tex]
Answer:
68,162 is the answer
Step-by-step explanation:
54394+ 13768 = 68162
What is the measure of 2x?
Answer:
130 °
....................
Hi! I'm happy to help!
To solve this, we need to first solve for x.
The total of all the interior angles in a quadrilateral (closed, not curved, shape with 4 sides) is 360°. So, we can single out the xs by subtracting the other angles.
360-89
271
271-76
195
Now that we know that, without the other two angles, the quadrilateral is worth 195. The only 2 angles left have x in them. Since there are 3 xs, we can use this to find out our x.
3x=195
Now, divide both sides by 3.
x=65
Now that we have the value of x, the measure of 2x will just be double that.
65×2
130
The measure of 2x is 130°
I hope this was helpful, keep learning! :D
11. The sales tax rate for the state of Washington was 8.4%. What is the final cost of a $5,700 car in Washington,
including tax?
$
Answer:
$5853.60
Step-by-step explanation:
8.4% of $5400 = $453.60
$453.60 + $5400 = $5853.60
Write a function rulle
Hey, there! Based on the table from the picture, every hour increased is $6 increased for the cost of the rental.
First hour: $10
Second hour: $16
Third hour: $22
Fourth hour: $28
So, final answer is:
+1 hour=+$6.
Anyways, I hope this helps! Enjoy your day! Take care!
What is 48% of 159 express your answer rounded correctly to one decimal place
Answer:
.48 * 159 = 76.32
Step-by-step explanation:
Benchmark of 6/10, how do I get this worked out
Answer:
0.60 or 60% is the answer!
Step-by-step explanation:
Hope I helped
calculate the amount that would remain in an investment account after 3 years if you were to deposit 1500 every month starting today but would withdraw 750 at the end of every 12 month period. interest is 14% compounded monthly
Answer:
58995
Step-by-step explanation:
Can you help me please??
Answer:
either a or c sry
Step-by-step explanation:
A straw is placed inside a rectangular box that is 6 inches by 8 inches by 5 inches, as shown. If the straw fits exactly into the box diagonally from the bottom left corner to the top right back corner, how long is the straw? Leave your answer in the simplest radical form.
Assume that when adults with smartphones are randomly selected , 44% use them in meetings or classes. If 8 adult smartphone users are randomly selected , find the probability that exactly 3 of them use their smartphone in meetings or classes
Answer:
you just answered your own question
Please help me out quickly!!
Answer:
third answer! x = -2/9
Step-by-step explanation:
:)
Which expression is equivalent to 8(6g)?
Answer:
8(6g) = 48g
I hope I helped you^_^
The solution to 12x = 36 is x =___. (Only input whole number) Numerical Answers Expected! Answer for Blank 1:
Hi ;-)
12x = 36
x = 36 : 12
x = 3
Hey there! I'm happy to help!
We see that an unknown number x multiplied by 12 is equal to 36. How do we find this number? Well, you can simply go backwards and divide 36 by 12, and this will give us our answer.
36÷12=3
Another way you can think about this is dividing both sides of the equation by 12, which isolates our x.
12x=36
12x÷12=36÷12
x=3
Have a wonderful day and keep on learning! :D
y link
MIT
MULTIPLE CHOICE QUESTION
Is the following statement, True or False?
lf this same substance were cooling, the
freezing point, or moment when the
substance begins to freeze would happen
at 25 degrees Celsius.
Answer:
False False False because freezing point is 100 degree celsius
find the length of BE BC=3x+47 DE=10 BD=x+27 CE=x+26
Answer:
B____C____D_____E
BC+ CE = BD + DE
(3x+47) + (x+26) = ( x+27) + (10)
4x + 73 = x + 37
4x – x = 37 – 73
3x = ‐ 36
x = – 36/ 3 —> x = – 12
BC = 3x + 47 = 3(-12) + 47 = - 36 + 47 = 11
BD = x+ 27 = –12+27 = 15
CE = x + 26 = –12+26= 14
So; BE = BD+ DE = 15+ 10= 25Or ;BE= BC + CE = 11+ 14 = 25I hope I helped you^_^
discoverings of marie curie
Answer:
Marie won the 1911 Nobel Prize in Chemistry for her discovery of the elements polonium and radium, using techniques she invented for isolating radioactive isotopes. Under her direction, the world's first studies were conducted into the treatment of neoplasms by the use of radioactive isotopes.
Basically, she discovered new elements and one of them is radiation which is used world-wide for cancer and other things
Step-by-step explanation:
find out the things that can be purchased between rupees 500 to rupees 100 for your house ?
We can buy small vases,Flower pots for our houses.
[tex]\sf\orange{Hope \: it \: helps \: please \: mark \: me \: as \: brainliest!}[/tex]
please solve its due
Answer:
2118+3391+4785+6354= 16648
Step-by-step explanation:
compute the absolute value and the conjugate of z = (1 + i) ^ 6
Answer:
z = (1 + i)6 =³√2(cos π4+ isinπ4)´6= 8 µcos3π2+ isin3π2)¶= −8i. Hence |z| = 8 and ¯z = 8i. z = (1 + i)
I think I am not real sure
In polar form, we have
[tex]1+i = \sqrt2 e^{i\pi/4}[/tex]
Then by DeMoivre's theorem,
[tex](1+i)^6 = \left(\sqrt2\right)^6 e^{i\,6\pi/4} = 2^3 e^{i\,3\pi/2} = -8i[/tex]
Then
[tex]|z| = |-8i| = \boxed{8}[/tex]
and
[tex]\bar z = \overline{-8i} = \boxed{8i}[/tex]
4. Solve the following equations and verify your answers: (i) 3 (x + 7) = 18
Answer:
8.33333333333
Step-by-step explanation:
Solution for 3x-7=18 equation:
3x - 7 = 18
3x = 18 + 7
3x = 25 (divide both sides by 3 to get x)
3x/3 = 25/3
x = 8.33333333333