Matrices & Determinants Questions and Answers

Compute all the minors and cofactors of 1 2 3 2 0 1` 2 3 4

Find 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.

Consider 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.

Given that f(x)=1x-4, the vertical asymptote(s) of f is: X=?

Describe 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.)

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

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

Find 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.

Solve 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

B) 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.

The 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.

Let (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)?

Thinking 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.

Let 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):

Austin 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.

Which of the 8 symmetries of a unit square are symmetries of a taxicab circle

If 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 ____________

Let 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?

-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

a = <-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.

Factor matrix A into E's so that A = E₁E₂...En A= 1 2 4 6

Let. 0 3 3 A= 1 1 0 -1 2 3 Assume that AB = A + 2B. Find the matrix B.

Find the optimal row and column strategies and the value of the matrix game. 5 -1 1 4 2 3 -2-3 1

Suppose, 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:

Solve 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)

Which 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

Use 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

Let 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.

The 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

Which 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

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

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

Consider 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]

The 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

Given 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

By 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.

SOLVE 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

A woman is m years old. How old will she be in ter years' time?

You 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?.

Select 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.

Graph the following function. y = 4 cos (x + π/8)

Use 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]