![Find The Inverse Of The Matrix [6, 7] [-6, 7] If It Exists](https://brainimage.s3.us-east-005.backblazeb2.com/contents/attachments/E1XI6hUieq9t0og42R3NCHu2XUZ2sQcN.png)
Answers
For any square matrix A, if it is invertible, its inverse is given by
[tex]A^{-1} = \dfrac1{\det(A)} \mathrm{adj}(A)[/tex]
where adj(A) is the so-called adjugate matrix, which is the transpose of the cofactor matrix of A. If you don't know what that is, that's not terribly important. What is important is that the inverse does not exist if the determinant, det(A), is zero.
We have
[tex]A=\begin{bmatrix}6&7\\-6&7\end{bmatrix} \implies \det(A) = 6\times7-7\times(-6)=84 \neq 0[/tex]
so the inverse does indeed exist. (And we eliminate D as an answer.)
From the given choices, it's quite clear that C must the correct answer, since we know det(A) = 84, and we can easily confirm that
[tex]\begin{bmatrix}7&-7\\6&6\end{bmatrix} \begin{bmatrix}6&7\\-6&7\end{bmatrix} = \begin{bmatrix}84&0\\0&84\end{bmatrix}[/tex]
so that multiplying by 1/84 recovers the identity matrix.
We are given with a square matrix and we are asked to find it's inverse if it exists . So , let's recall some important points first :-
Inverse can only be found of square matrices.If [tex]{\bf A}[/tex] is a square matrix than , it's inverse is denoted by [tex]{\bf A^{-1}}[/tex] , and is given by [tex]{\bf{A^{-1}=\dfrac{1}{adj(A)}det(A)}}[/tex] , where adj(A) is the adjoint of the matrix A and det(A) is the determinant of the matrix AInverse of a matrix exist if A is non-singular Non-Singular means that det(A) ≠ 0 Adjoint of a matrix is the matrix of transpose of cofactors Transpose of a matrix is founded by exchanging it's rows by columns and columns by rows and is denoted by [tex]{\bf{A^{T}}}[/tex]Now , in this question let's assume that [tex]{\sf A=\begin{bmatrix}6 & 7 \\ -6 & 7\end{bmatrix}}[/tex]
Now , Calculating det(A) :-
[tex]{:\implies \quad \sf det(A)=\begin{vmatrix}6 & 7 \\ -6 & 7 \end{vmatrix}}[/tex]
[tex]{:\implies \quad \sf det(A)=42-(-42)}[/tex]
[tex]{:\implies \quad \sf det(A)=42+42}[/tex]
[tex]{:\implies \quad \sf det(A)=84}[/tex]
As , det(A) ≠ 0 . So , [tex]{\bf A^{-1}}[/tex] exists . Now , we need to find the matrix of cofactors first , but let's find cofactors first , so here ;
[tex]{\blacktriangleright \sf C_{11}=7,\: C_{12}=6,\: C_{21}=-7,\: C_{22}=6}[/tex]
Now , let's assume that matrix of cofactors is C , so putting the cofactors as elements of the matrix , C will be ;
[tex]{:\implies \quad \sf C=\begin{bmatrix}7 & 6 \\ -7 & 6\end{bmatrix}}[/tex]
Now , adj(A) will be found by interchanging it's rows by columns and vice versa.
[tex]{:\implies \quad \sf adj(A)=\begin{bmatrix}7 & -7 \\ 6 & 6\end{bmatrix}}[/tex]
Now as [tex]{\sf A^{-1}}[/tex] is given by [tex]{\bf{A^{-1}=\dfrac{1}{adj(A)}det(A)}}[/tex]
[tex]{:\implies \quad \bf \therefore \quad \underline{\underline{A^{-1}=\dfrac{1}{84}\begin{bmatrix}7 & -7 \\ 6 & 6\end{bmatrix}}}}[/tex]
Hence , Option C) [tex]{\sf \dfrac{1}{84}\begin{bmatrix}7 & -7 \\ 6 & 6\end{bmatrix}}[/tex] is correct :D
Related Questions
Ebony put 10 cubes into two stacks. One stack has 4 more cubes than than the other stack. How many cubes are in each stack? Show your work.
Answers
Answer:
Step-by-step explanation:
10 cubes ÷2 stacks = 5, so in each stack 5.
one stack has 5
Second stack has 4 more cubes than the other. (5+4=9)
So the answer is 5 and 9
The number of cubes is 3 and 7 in two stacks consisting of 10 stacks total.
Suppose Ebony has 'x' cubes in one stack
So, the number of cubes in another stack =x+4
What is a cube?A cube is a solid having all three dimensions i.e. length, breadth, and height equal to each other.
According to the question,
Total number of cubes = 10
x+(x+4) = 10
2x+4 =10
2x=6
x=3
So, Ebony has 3 cubes in one stack
Number of cubes in another stack =3+4 =7cubes
Therefore, The number of cubes is 3 and 7 in two stacks consisting of 10 stacks total.
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Find the inverse function of f informally.
f(x) = x^3 + 8
Answers
Answer:
[tex]f^-1 (x) = \sqrt[3]{x - 8}[/tex]
Step-by-step explanation:
Custom file created rather than math ML for neatness.
Once again- help pls T-T
Answers
Answer:
Example 1) sin(42) ×25=16.73
Example 2) 12÷tan(75)=3.22
Example 3) 19÷cos(36)=23.49
Example 4) tan(53)×7=9.29
Using Postulates and/or Theorems learned in Unit 1, determine whether ABC~AXY.
Show all your work and explain why the triangles are similar or why they are not.
Answers
Explanation:
AB = AX+XB = 8+22 = 30
The ratio of AB to AX is AB/AX = 30/8 = 3.75
AC = AY+YC = 9+33 = 42
The ratio of AC to AY is AC/AY = 42/9 = 4.67 approximately
Since we don't get the same result, this means that the equation AB/AX = AC/AY is not true. The sides are not in proportion with one another, and therefore the triangles aren't similar.
The sides need to be in proportion with one another so we could use the SAS similarity rule.
Considering the SAS similarity theorem, triangles ABC and AXY are not similar triangles.
What is the SAS Similarity Theorem?The SAS Similarity postulate or theorem, states that, when two triangles have two pairs of corresponding sides whose ratios are the same, and also has a pair congruent included angles, then, both triangles are similar.
Triangles ABC and AXY have a pair of congruent included angles, ∠BAC ≅ XAY.
Let's find out if their corresponding sides have equal ratios.
Thus:
AB/AX = 30/8 = 3.75
AC/AY = 42/9 = 4.67
Their ratios are not equal.
Therefore, considering the SAS similarity theorem, triangles ABC and AXY are not similar triangles.
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Solve each real-world problem below, showing all of your work. You must state the starting amount and rate of change for each problem. You must also write the linear equation used to solve the problem. The memory card in Carla's camera can hold 1,024 pictures. On April 30th, Carla sees that she has 235 pictures on the memory card. She plans to take 35 pictures each day in the month of May. How many pictures will Carla have on the memory card on May 16th? On what day will the memory card be full?
Answers
Carla would have about 795 pictures on May 16th.
Linear equationA linear equation is in the form:
y = mx + b
where y,x are variables, m is the rate of change and b is the y intercept.
Let y represent the number of pictures after x days.
At April 30th, Carla sees that she has 235 pictures, hence b = 235. Also, she takes 35 pictures each day, hence m = 35. The equation is:
y = 35x + 235
On May 16th, x = 16 hence:
y = 35(16) + 235 = 795
Carla would have about 795 pictures on May 16th.
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a standard deck of cards missing the queen of hearts in the 2 of clubs what is the probability of pulling either an ace or a spade
Answers
=======================================================
Explanation:
In a standard deck, there are 52 cards.
If this deck is missing the queen of hearts and 2 of clubs, then we really have 52-2 = 50 cards in the deck.
There are 4 aces and 13 spades. Those values add to 4+13 = 17, but we need to subtract off 1 to account for the ace of spades counted twice. We have 17-1 = 16 cards that are either an ace, a spade, or both.
Or you can think of it like saying 13 spades + 1 ace of hearts + 1 ace of diamonds + 1 ace of clubs = 16 cards total.
-----------------
The event space has A = 16 cards in it, while the sample space has B = 50 cards.
The probability we're after is A/B = 16/50 = 8/25
Shade (A U C) intersection B’
Answers
There is a quadrilateral ABCD in which Q is a point on the side AD and the length of side AB is 18. Segment BQ is the angle bisector of angle ABC and segment CQ is the angle bisector of angle BCD.
Quadrilateral ABCD is a parallelogram. What is BC?
BC =
Answers
Answer:
ABCD is the quadrilateral
Write an expression for each of the expressions below by combining the like terms: 2y + 7y
Answers
9y
(space fillerrrrr)
Please help me. Thank you so much. :3
The fractions 810 and 58 are graphed on these number lines.
How do the fractions compare?
Two number lines lined up vertically. The first number line goes from 0 to 1 and is divided into tenths. There is a dot at 8 tenths. The second number line goes from 0 to 1 and is divided into eighths. There is a dot at 5 eighths.
Drag and drop the symbol into the box to correctly compare the fractions.
810 Response area 58
Answers
Answer:
8/10 > 5/8
Step-by-step explanation:
8/10 > 5/8
Hope this helps!
When a cube with sides numbered 1 trough 6 is
rolled 1 time what is the probability of rolling a 4
or greater
Answers
Answer:
The probability of rolling a 4 or greater is 12
Step-by-step explanation:
hope it helps you
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Martha simplified the expression ———-, putting it into the form ———-. what is the value of a - b?
The expression is in the photo.
A. 2
B. 4
C. 5
d. 7
Answers
2x^2- 7x + 13
a-b
7x+13
2) Suppose 3 people pooled their money. How much money did each winner receive?A) $120B) $140C) $160D) $480
Answers
Зn - 4 = 20
Evaluate the equation
Answers
Move -4 to the right (it becomes a +4 now)
3n = 20 + 4
3n = 24
Divide both sides by 3
n= 8
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What is the length of the hypotenuse of the right triangle shown below? Type your answer as an integer. A right triangle has two sides of length 3 and 4.
Answers
Answer:
5
Step-by-step explanation:
This is the Pythagorean theorem:
[tex]a^{2} +b^{2} =c^{2}[/tex]
The hypotenuse is c so we need to make c the subject:
[tex]c = \sqrt{a^{2}+b^{2} }[/tex]
To do this I simply square rooted both sides.
Now we substitute values given to get the answer of the hypotenuse:
[tex]c = \sqrt{3^{2} + 4^{2} }[/tex]
[tex]c = \sqrt{9+16}[/tex]
[tex]c = \sqrt{25}[/tex]
c=5
The lengths 3,4,5 of a triangle are known as a Pythagorean triple.
A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c².
What's the area of a triangle whose base has a length of 27 units and whose height is
16 units?
A) 108 square units
B) 144 square units
C) 216 square units
OD) 432 square units
Answers
Answer:
216 units^2
Step-by-step explanation:
the formula to calculate area of triangle is 1/2 b×h
so...
[tex] \frac{1}{2} (27) \times (16) \\ = 216 {units}^{2} [/tex]
Answer:
Step-by-step explanation:
144 units^3
PLEASE HELP ME I WILL GIVE BRAINLEST
Answers
Evaluate the given functions at the indicated value and simplify.
Answers
Answer:
See Below
Step-by-step explanation:
For each of these, you just replace x with what's in the parenthesis.
f(-2) = 4(-2)^2 - 2(-2) + 1
f(-2) = 16 + 4 + 1
f(-2) = 21
g(5) = [tex]\sqrt{4(5) - 16}[/tex]
g(5) = [tex]\sqrt{20-16}[/tex]
g(5) = [tex]\sqrt{4}[/tex]
g(5) = 2
f(3r) = 4(3r)^2 - 2(3r) + 1
f(3r) = 4(9r^2) - 6r + 1
f(3r) = 36r^2 - 6r + 1
f(a+b) = 4(a+b)^2 - 2(a+b) +1
f(a+b) = 4 (a^2 + 2ab + b^2) - 2a - 2b + 1
f(a+b) = 4a^2 + 8ab + 4b^2 - 2a - 2b + 1
The terminal side of an angle θ in standard position is in quadrant 1. If sinθ=0.8, find cosθ.
Answers
Answer:
[tex]cos\theta=\frac{3}{5}[/tex]
Step-by-step explanation:
Since [tex]sin\theta=0.8[/tex] is the same as [tex]sin\theta=\frac{4}{5}[/tex] and [tex]sin\theta=\frac{opposite}{hypotenuse}[/tex], then we know that the side opposite to the angle [tex]\theta[/tex] is 4 units and the hypotenuse is 5 units.
Because [tex]cos\theta=\frac{adjacent}{hypotenuse}[/tex], then we have to find the adjacent side to the angle [tex]\theta[/tex] using the Pythagorean Theorem:
[tex]a^2+b^2=c^2\\\\(adjacent)^2+(opposite)^2=(hypotenuse)^2\\\\(adjacent)^2+4^2=5^2\\\\(adjacent)^2+16=25\\\\(adjacent)^2=9\\\\adjacent=3[/tex]
Therefore, since the adjacent side is 3 units, then [tex]cos\theta=\frac{3}{5}[/tex]
Add the fraction and whole numbers
5 13/16 +
2 1/8 +
3 3/4
Answers
Answer:
The sum of the three mixed numbers is 11 and 9.5/16.
Step-by-step explanation:
5 and 13/16
2 and 1/8 = 2 and 0.5/16
3 and 3/4 = 3 and 12/16
10 and 25.5/16
11 and 9.5/16
Answer:
[tex]11\frac{11}{16\\} \\[/tex]
Step-by-step explanation:
Separate fractions and whole numbers
5 2 3 13/16 1/8 3/4
Add whole numbers and fractions (where the denominator is 16)
5+2+3=1
13/16+1/8+3/4 =
13/16+2/16+12/6=
27/16 =
[tex]1\frac{11}{16\\} \\[/tex]
27/16+10 = [tex]11\frac{11}{16\\}[/tex]
Simplify the following expression. square root x^5y^5 times 4 square root 5x^13y^12
Answers
Answer:
[tex]4x^{9} y^{8} \sqrt{5y}[/tex]
Step-by-step explanation:
HELP IM DESPRITE
Matthew can spend up to $75 on supplies for his garden. The flowers will cost $38 and he wants to spend the rest on bags of top soil. Each bag of top soil costs $6.
How many bags of top soil can he buy?
Answers
Answer:
6 bags.
Step-by-step explanation:
Subtract 38 from 75 to represent the remaining amount of Matthew's money.
[tex]75-38=37.[/tex]
Matthew has $37 left over from buying flowers.
Divide 37 by 6 to represent how many bags of soil he can buy.
[tex]37\div6=6\frac{1}{6}[/tex]
Matthew can only buy 6 bags of soil, 1/6 isn't included because you can't buy a fraction of a bag.
PLEASE HELP ME QUICKKK, FIRST CORRECT PERSON GETS BRAINLIEST
Answers
Answer:
C
Step-by-step explanation:
hope this helps :)
Find the x- and y -intercepts of the graph of the linear equation -6x+9y=-18 .
The x -intercept is
.
The y -intercept is
.
Answers
y intercept= (0,-2)
Answer:
The x -intercept is (3, 0)
The y -intercept is (0, - 2)
Step-by-step explanation:
Given equation is: -6x + 9y = -18
To find x - intercept:
Plug y = 0 in the given equation.
-6x + 9(0) = - 18
-6x + 0 = -18
-6x = - 18
x = (-18)/(-6)
x = 3
The x -intercept is (3, 0)
To find y - intercept:
Plug x = 0 in the given equation.
-6(0) + 9y= - 18
0 + 9y = -18
9y = - 18
y = (-18)/(9)
y = - 2
The y -intercept is (0, - 2)
I don't know how to do this
Answers
Answer:
a) 2x + 3; b) 0
Step-by-step explanation:
Area = [tex]4\pi x^2+12\pi x+9\pi =\pi (4x^2+12x+9)=\pi (2x+3)^2[/tex]
Area = [tex]\pi r^2[/tex]
a) radius r = 2x + 3
b) 2x + 3 > 0
x > -3/2
The least possible integer value of x is 0
Type an order pair ()
Answers
To get P2, we will do the same to the midpoint
P2 = (-1+2, 6+2) = (1,8)
Check whether this is a quadratic equation or not ...
=>>>>> x²-2x = (-2)(3-x)
Answers
Answer:
yup it is ...
x²-2x = -6+2x
=> x²-4x +6
By expanding the equation, we can see that it is quadratic.
Is the equation quadratic?
A quadratic equation is any polynomial of degree 2.
Here we have:
x^2 - 2x = (-2)*(3 - x)
Expanding this we get:
x^2 - 2x = -6 + 2x
Moving all to the same side:
x^2 - 2x - 2x + 6 = 0
x^2 - 4x + 6 = 0
This is a polynomial of degree 2, so this is a quadratic equation.
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[tex]5^{3x}-5^{3x-1}=4[/tex] In the following equation, solve for x (to two decimal places if necessary) and write all the steps until the final answer.
Answers
Answer:
[tex]x = \frac{1}{3} [/tex]
or x = 0.33
Step-by-step explanation:
[tex] {5}^{3x} - {5}^{3x - 1} = 4 \\ \\ \implies{5}^{3x} - {5}^{3x} . {5}^{ - 1} = 4 \\ \\ \implies{5}^{3x}(1 - {5}^{ - 1} )= 4 \\ \\ \implies{5}^{3x} \bigg(1 - \frac{1}{5} \bigg)= 4 \\ \\ \implies{5}^{3x} \bigg(\frac{5 - 1}{5} \bigg)= 4 \\ \\ \implies{5}^{3x} \times \frac{4}{5} = 4 \\ \\ \implies{5}^{3x} = 4 \times\frac{5}{4} \\ \\ \implies{5}^{3x} =5 \\ \\ \implies \: 3x = 1 \\ \\ \huge \implies \: x = \frac{1}{3} \\\\ \huge \implies \: x = 0.33[/tex]
i need a help pleass ...................
Answers
Answer:
Yes, it is a right triangle. The distance is 6,2.
Step-by-step explanation:
The square of 15 is 225, and the square of 12 is 144. The square of 9 is 81. 144 + 81 = 225, so it's a right triangle.
you are required to read any
four books from a list of fifteen. How
many different selections are possible?