Matrices & Determinants Questions and Answers


Algebra
Matrices & DeterminantsFind the values of x, y and Z such the matrix below is skew symmetric.
0 X 3
2 y -1
Z 1 0
• Give an example of a symmetric and a skew
symmetric 3 by 3 matrix.
•Prove that A² is symmetric whenever A is skew
symmetric.
• Determine an expression for det(A) in terms of
det(AT) if A is a square skew symmetric.
•Assume that A is an odd order skew symmetric
matrix. Prove that det(A) is an odd function
in this case.

Algebra
Matrices & DeterminantsConsider the equations
5x₁ + x₂ + 3x₃ + 6 = 0
-5x₁-2x₃ +7= 0.
Apply Gaussian elimination to convert this system into (row) echelon form.
Find the general solution and write it as a line or plane in parametric form.


Algebra
Matrices & DeterminantsDescribe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix.
1 4 -2 5
0 1 -5 5
x= x₂_ + x₁__
(Type an integer or fraction for each matrix element.)

Algebra
Matrices & DeterminantsSolving the following system of equations using any method. Show each step clearly.
X + 2Y + 4Z = 11
2X + Y + 2Z = 7
3X - Y - 2Z = -2

Algebra
Matrices & DeterminantsPrecalculus
Find the vertex, focus, and directrix of the parabola. Graph the equation.
2y²+8y-4x+6=0

Algebra
Matrices & DeterminantsFind the vector equation of the line through the point (-4,7,-2) parallel to the vector [3,5,-5).
You may use x to abbreviate the general position vector [x. y, z). Both the row vector notation or i, j, k notation of vectors (used in previous questions) may be used. Any single lowercase letter except i, j, k, x,r, y or 2 may be used as a parametric variable, e.g. is a nice choice.

Algebra
Matrices & DeterminantsSolve each initial-value problem.
y" + 2y' + 5y = g(t); y(0) = 1, y'(0) = 2
y" - y' - 6y = {t, if t < 2;
{1, if t > 2; y(0) = 0, y'(0) = 0

Algebra
Matrices & DeterminantsB) Please explain sine, cosine and tangent in the context of angles on the coordinate plane. Again, describe the ratios and what they mean. Please use at least 3 sentences.
C) Please explain how we connected our understanding of sine, cosine and tangent in right triangles to develop an understanding of sine, cosine and tangent in the coordinate plane. Please use at least 2 sentences.

Algebra
Matrices & DeterminantsThe histogram below shows the scores for Mrs. Smith's first block class at Rock Middle School. If an 85 is the lowest score a student can earn to receive a B, how many students
received at least a B?
A) 4
C) 6
B) 10
D) 15
![Determine the eigenvalues of the matrix [3 -3] . Find at least one corresponding eigenvector from the eigenvalue that you have determined.
*you may attach your scanned solution or input your final answer for the eigenvalues, and the corresponding eigenvector.](https://media.kunduz.com/media/sug-question/raw/80455109-1659817733.5498438.jpeg?w=256)
Algebra
Matrices & DeterminantsDetermine the eigenvalues of the matrix [3 -3] . Find at least one corresponding eigenvector from the eigenvalue that you have determined.
*you may attach your scanned solution or input your final answer for the eigenvalues, and the corresponding eigenvector.

Algebra
Matrices & DeterminantsLet (V, (,)) be an inner product space and let f, g be vectors in V. If
||f|| = 4, ||g||= 5 and || f + g = √31, then what is the value of (f, g)?

Algebra
Matrices & DeterminantsThinking
Find AN unit vector that is perpendicular to the yz-plane and perpendicular to the
-2i+7j - 3k
vector
Keep your answer in exact form.
please show all work. thanks.

Algebra
Matrices & DeterminantsLet A be a 3 x 8, B be a 8 x 7 and C be a 7 x 3 matrix. Determine the size of the following matrices (if they do not exist, type N in both answer boxes):

Algebra
Matrices & DeterminantsAustin is going to invest in an account paying an interest rate of 2.2% compounded daily. How much would Austin need to invest, to the nearest hundred dollars, for the value of the account to reach $27,000 in 6 years?
![Let T: M₂₂→ M₂₂ be defined by
T([a b]) = [ 2c a+c]
c d b-2c d
(a) Find the eigenvalues of T.
(b) Find bases for the eigenspaces of T.](https://media.kunduz.com/media/sug-question/raw/80463728-1659817233.6647193.jpeg?w=256)
Algebra
Matrices & DeterminantsLet T: M₂₂→ M₂₂ be defined by
T([a b]) = [ 2c a+c]
c d b-2c d
(a) Find the eigenvalues of T.
(b) Find bases for the eigenspaces of T.

Algebra
Matrices & DeterminantsWhich of the 8 symmetries of a unit square are symmetries of a taxicab circle

Algebra
Matrices & DeterminantsIf θ=5π/2, then find exact values for the following. If the trigonometric function is undefined for θ=5π/2, enter DNE.
A. sec (0) equals ____________
B. csc (0) equals ____________
C. tan (0) equals ____________
D. cot (0) equals ____________

Algebra
Matrices & DeterminantsLet A be an n x n matrix where n is odd and such that A = -A^T.
(a) Show that det(A) = 0.
(b) Does this remain true in the case n is even?

Algebra
Matrices & Determinants-1 2 0 4
1 4 1 -3
A= -1 1 0 1
-3 8 1 3
Find the inverse A-¹ of the given matrix A if it exists.
1. Construct the augmented matrix
2. Show all steps to modify it to row echelon form applying equivalent elementary row operations
Show all steps to modify it to reduced row echelon form applying equivalent elementary row
operations
3.Verify your solution
4.Show the answer

Algebra
Matrices & Determinantsa = <-3,-5> and b = <1,4>.
Represent a + b using the parallelogram method.
Use the Vector tool to draw the vectors, complete the parallelogram method, and draw a + b.
To use the Vector tool, select the initial point and then the terminal point.
![Which of the following statements are true?
A. If the columns of an m X n matrix A span Rᵐ, then the equation Ax=b is consistent for each b in Rᵐ.
B. Every matrix equation Ax=b corresponds to a vector equation with the same solution set.
C. The equation Ax=b is referred to as a vector equation.
D. A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution.
E. If the augmented matrix [ A b] has a pivot position in every row, then the equation Ax=b is inconsistent.
F. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.](https://media.kunduz.com/media/sug-question/raw/80538108-1659816460.1988146.jpeg?w=256)
Algebra
Matrices & DeterminantsWhich of the following statements are true?
A. If the columns of an m X n matrix A span Rᵐ, then the equation Ax=b is consistent for each b in Rᵐ.
B. Every matrix equation Ax=b corresponds to a vector equation with the same solution set.
C. The equation Ax=b is referred to as a vector equation.
D. A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution.
E. If the augmented matrix [ A b] has a pivot position in every row, then the equation Ax=b is inconsistent.
F. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.


Algebra
Matrices & DeterminantsLet. 0 3 3
A= 1 1 0
-1 2 3
Assume that AB = A + 2B. Find the matrix B.

Algebra
Matrices & DeterminantsFind the optimal row and column strategies and the value of the matrix game.
5 -1 1
4 2 3
-2-3 1

Algebra
Matrices & DeterminantsSuppose, a = 10i +14 j and b = ki +17 j
9.1) Find the exact value of k such that a and b are parallel.
Answer:
You have not attempted this yet
9.2) Find the exact value of k such that a and b are perpendicular.
Answer:

Algebra
Matrices & DeterminantsSolve for X in the equation 3X + A = B, where
Find the focus and directrix of the parabola y=-2x². Then sketch the parabola
Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.
Vertices: (±5,0); foci: (±2,0)

Algebra
Matrices & DeterminantsWhich of the following is always equal to the inverse of (AB) ?
(A) B-¹ + A-¹
(B) B-¹A-¹
(C) A-¹ + B-1
(D) A-¹B-1

Algebra
Matrices & DeterminantsUse the table below to find:
(fog)(5) =
(gof)(8) =
(fof)(-2)=
(gog)(0)=
x -2 0 4 5 7 8 10 13
f(x) 5 10 0 8 -2 4 13 7
g(x) 8 5 4 10 -2 7 13 0

Algebra
Matrices & DeterminantsLet F = (4z+4x³)i + (4y + 4z + 4 sin(y³))j+ (4x + 4y + 4eˣ³) k
(a) Find curl F.
(b) What does your answer to part (a) tell you about ∫F.dr where C is the circle
c
(x-10)²+(y-25)²=1 in the xy-plane, oriented clockwise?
(c) If C is any closed curve,what can you say about∫ F.dr?
c
(d) Now let C be the half circle (x-10)² + (y-25)²=1 in the xy-plane with y> 25, traversed from (11, 25) to (9, 25). Find ∫F .dr by using your result from (c) and
c
considering C plus the line segment connecting the endpoints of C.
![Show that the columns of the 3 by 4 matrix A [with rows] = (1 1 1 1) (1 2 −1 3) (1 0 3 −2) ∈ M₃₄(Q) span Q³. Then find a rational 4x3 matrix B such that AB=I₃.](https://media.kunduz.com/media/sug-question/raw/80814214-1659815340.5762067.jpeg?w=256)
Algebra
Matrices & DeterminantsShow that the columns of the 3 by 4 matrix A [with rows] = (1 1 1 1) (1 2 −1 3) (1 0 3 −2) ∈ M₃₄(Q) span Q³. Then find a rational 4x3 matrix B such that AB=I₃.

Algebra
Matrices & DeterminantsThe matrix -3 3
A = 5 5
has a singular value decomposition A = UΣVᵗ where
U=0 -1
-1 0
and Σ= 5√2 0
0 5√2
and V=-1/√2 1/√2
-1/√2 -1/√2
The 2,2 entry of the pseudoinverse A+ is(A+)₂₂=1/z
What is z?
![Let us consider the following matrices.
Calculate the following. If a result does not exist, explain why.
[a] AB
[b] AC
[c] BC
[d] CB
[e] BD
[f] CD](https://media.kunduz.com/media/sug-question/raw/80595966-1659815090.6407888.jpeg?w=256)
Algebra
Matrices & DeterminantsLet us consider the following matrices.
Calculate the following. If a result does not exist, explain why.
[a] AB
[b] AC
[c] BC
[d] CB
[e] BD
[f] CD

Algebra
Matrices & DeterminantsWhich of the following pairs of vectors is a pair of independ
2 -5
-1 -2
Select one:
A {(1,2),(-5,1)}
B {(1,1),(-5,1)}
C. {(1,1), (2,1)}
D. {(2,1),(-5,1)}
![Let
A = [-1 1]
[1 1]
[1 -2]
Which of the following vectors are not in the subspace {As : s € R²}?
□ [6 2 -10]ᵀ
□ [1 1 1]ᵀ
□ [0 -1 2]ᵀ
□ [0 -2 1]ᵀ](https://media.kunduz.com/media/sug-question/raw/80654996-1659815043.339204.jpeg?w=256)
Algebra
Matrices & DeterminantsLet
A = [-1 1]
[1 1]
[1 -2]
Which of the following vectors are not in the subspace {As : s € R²}?
□ [6 2 -10]ᵀ
□ [1 1 1]ᵀ
□ [0 -1 2]ᵀ
□ [0 -2 1]ᵀ

Algebra
Matrices & DeterminantsSketch several periods of
f(x) = sin(πx) within -1/2<x<1/2
and expand it in an appropriate Fourier series.

Algebra
Matrices & DeterminantsIf possible, find AB, BA, and A². (If not possible, enter IMPOSSIBLE.)
A = 1 4 , B= 3 -1
2 4 -1 6

Algebra
Matrices & DeterminantsConsider the system
x₁ = x₂ + sin x₁
x₂ = θx² + u
y = x1
where the parameter is not exactly known but can be bounded as follows:
|θ| ≤ 2
Design a sliding surface controller to stabilize the origin in the presence of the uncertainty in parameter θ.
![Find A⁵ and C⁻³ by using Eigenvalues and Eigen-vectors:
A=[1/-2+1/4] C=[1/1/4 +2/0/-4 -1/1/5]](https://media.kunduz.com/media/sug-question/raw/80851169-1659814488.4477992.jpeg?w=256)
Algebra
Matrices & DeterminantsFind A⁵ and C⁻³ by using Eigenvalues and Eigen-vectors:
A=[1/-2+1/4] C=[1/1/4 +2/0/-4 -1/1/5]

Algebra
Matrices & DeterminantsThe set B=(1-t²,2t-t²-t-t²) is a basis for P₂. Find the coordinate vector of p(t)=-1+13t-6t² relative to B.
(P)B= 4 5 -3

Algebra
Matrices & DeterminantsGiven a system of two linear equations into variables, find the solution of the system algebraically and graphically. Solve by graphing, substitution method, and elimination method.
4x +3y=7
x-2y = -1

Algebra
Matrices & DeterminantsBy using de Moivre's Theorem, prove that
cos5θ = 16 cos⁵θ - 20cos³θ + 5 cosθ
Hence find all the roots of the equation
48x⁵ - 60x³ + 15x + 2 = 0.

Algebra
Matrices & DeterminantsSOLVE Linear Regression using quadratic nonlinear polynomial regressing model,
then find the RS, RT, R² and R.
n=7 Σx₁y₁ = 1 ΣX²i= 119.5
ΣXi= 140 x=28/7=4
Σyi=24 y=24/7== 3.428571


Algebra
Matrices & DeterminantsYou are shown a graph of two lines that intersect once at the point equation, (-3/7, 7/3) what do you know must be true of the system of equations?.

Algebra
Matrices & DeterminantsSelect the correct choice below and fill in the answer boxes within the choice.
A. There are 2 possible solutions for the triangle.
The measurements for the solution with the longer side c are as follows.
m∠B=
m∠C = °
The length of side c =
The measurements for the solution with the shorter
side c are as follows.
m∠B=
m∠C=⁰
The length of side c =
B. There is only 1 possible solution for the triangle.
The measurements for the remaining angles B and C and side c are as follows
m∠B=
.m∠C=
The length of side c =
C. There are no possible solutions for this triangle.


Algebra
Matrices & DeterminantsUse the definition of a logarithm to rewrite log₆36 = x in exponential form. Do not solve.
![Write "T" or "F" to indicate true or false if vector u is a linear combination of the vectors in the set S. If υ can be expressed as a linear combination, write out the corresponding linear combination; otherwise, place 0 in all corresponding boxes.
_________: υ=[11/-3/5/20]
is a linear combination of the set S containing the vectors
[2/-5/4/-3] [-5/3/-5/-5] [4/-5/4/2]
The corresponding linear combination would be:
[11/-3/5/20]=______[2/-5/4/-3]+ [-5/3/-5/-5]+ [4/-5/4/2]](https://media.kunduz.com/media/sug-question/raw/80939443-1659812474.18862.jpeg?w=256)
Algebra
Matrices & DeterminantsWrite "T" or "F" to indicate true or false if vector u is a linear combination of the vectors in the set S. If υ can be expressed as a linear combination, write out the corresponding linear combination; otherwise, place 0 in all corresponding boxes.
_________: υ=[11/-3/5/20]
is a linear combination of the set S containing the vectors
[2/-5/4/-3] [-5/3/-5/-5] [4/-5/4/2]
The corresponding linear combination would be:
[11/-3/5/20]=______[2/-5/4/-3]+ [-5/3/-5/-5]+ [4/-5/4/2]