Matrices & Determinants Questions and Answers

Consider the following 3 x 3 matrix: [ζ 1 7 ζ 3 2 4 -1 1] Find the values of for which the matrix is: 1. positive definite 2. positive semidefinite 3. negative definite 4. negative semidefinite 5. indefinite

Use the inner product < p,q>= p(-1)q(−1) +p(0)q(0) + p(3)q(3) in P₃ to find the orthogonal projection of p(x) = 4x² + 5x - 7 onto the line L spanned by q(x) = 4x²-3x - 2. projL(p) =

Given A= [-2 -3] and B= [5 2] [5 2] [1 -1] use the Frobenius inner product and the corresponding induced norm to determine the value of each of the following: (A, B): =_____ ||A||F= _________ ||B||F= ___________- θA,B=______radians.

if A is a regular matrix, then A+Aʰ is positive definite

Write the eigen values and eigen vectors corresponding to the linear map represented by the following matrix. A = 3 0 8 -1

4. Determine whether the given matrix is invertible without computing its determinant. -1 -3 0 1 3 5 8 -3 -2 -6 3 2 0 -1 2 1

Show that every group of order 567 has a normal subgroup of order 27.

Let P : R³ → R³ be the skew-projection onto the plane II = {-(4x + y + 3z) = 0} parallel to the vector (1, −2, –4). Find bases for the 0 and 1 eigenspaces. Basis for V0= Basis for V₁=

Let A = 0 -1 2 2 Find PLU factorization. -1 3 4 1 2 -4 -3 12

Use Gauss Jordan to find the solution of system of Linear equations. Use elementary row operations in your solution. x -2y+ z = -1 x + 3z=-11 -4x + 7y + 4z = 3

Consider 3x + 3y + 3z=- 3k² 4y + 8z= -4 x + 4y +7z = 4k a) Using Gaussian elimination method find all value(s) of k such that the system becomes consistent. b) Using part (a), write down the solution(s) of the system if k = -1.

3) Find the values of x and y using pseudoinverse. X + 3y =17 5x + 7y =19 11x + 13y =23

Plot the ordered pair (1,-1). State in which quadrant or on which axis the point lies. Plot the ordered pair on the graph to the right. In which quadrant, or on which axis, does the point lie? I III II on the x-axis on the y-axis IV

Identify the intercepts. What are the intercepts? (Type an ordered pair. Use a comma to separate answers as needed.)

Solve the system of linear equations by graphing. y = 4x 3x-y=1 Use the graphing tool to graph the system.

Confirm that B is an orthonormal set, and find the projection of v onto span B B= 1/√6 -1 , 1/√3 1 and v= 8 1 -1 4 2 1 5

Which of the following was NOT an aspect of Taoism practiced in China? A. Cultivation of chi B. Use of alchemy C. Respecting social relationships D. Ancestor worship

Use the Gram-Schmidt process to find orthonormal bases for the spaces

3. Complete the following questions using the following planes T₁: 2x + 4y-6z -2 = 0 T₂: 7x+14y-21z-7=0 R3: 4x+8y-11z +5=0 a) Determine the normal vectors for each of the three planes [1K] b) Are any of the planes parallel? Show your work (If yes, determine the scalar multiple. If no, show that they are not multiples) [27] c) Are the normals for the three planes coplanar? [2A]

Find the 3x3 matrix that produces the described composite 2D transformation below, using homogeneous coordinates. Rotate points through 45° about the point (6,9).

Consider T(x,y) = (2x-y, -4x+2y), is T invertible? Why or why not? If so, compute the inverse, T-1, and compute the image of 8 under the inverse map. ie. Compute T^-1

6-Which of the following statements is true (a) det A = det A-¹ (b)det(A + B)=det(A)+det(B)

Perform the indicated operation, if possible. [4 8 5 -9 + [ 7 1 -11 3 6 -1 12 1] - 3 8-10 9] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The resulting matrix is B. The matrices cannot be added.

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these products exist. A is 2 x 5, and B is 1 x 2. Find the size of the product AB. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The size of product AB is B. The product AB does not exist. Find the size of the product BA. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The size of product BA is B. The product BA does not exist.

Solve the following system of equations. Let z be the parameter. 3x+2y-z=2 4x + 3y +z = 5 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. There is one solution, .. B. There are infinitely many solutions. The general solution is (z), where z is any real number. C. There is no solution.

To find the product matrix AB, the number of of A must be the same as the number of of B. To find the product matrix AB, the number of of A must be the same as the number of of B.

Use the echelon method to solve the system of two equations in two unknowns. Check your answers. 6x - 5y = -9 8x - y = 5 Select the correct choice below and fill in any answer boxes within A. The solution of the system is B. There are infinitely many solutions. The solution is (y), where y is any real number. C. There is no solution.

Use the Gauss-Jordan method to solve the system of equations. y=x-1 y=5+z z=0-x Select the correct choice below and fill in any answer boxes within your choice. A. There is one solution. The solution is (..), in the order x, y, z. B. There are infinitely many solutions. The solution is z), where z is any real number. C. There is no solution.

Fill in the blank with the correct answer that completes the following sentence. on a matrix correspond to transformations of a system of equations. on a matrix correspond to transformations of a system of equations.

Solve the following equations: a) tan (x-72) = -1 on the interval (-∞,∞). b) 2cos (x/3) + 1 = 0 on the interval [0,27). c) 2cos^2(x) -3cos(x) +1 = 0 on the interval [0,2π)

Gonzalez Manufacturing borrowed $24000. Part of the money was borrowed at 8%, part at 10%, and part at 12%. The annual interest was $2480, and the total amount borrowed at 8% and 10% was twice the amount borrowed at 12%. Use Gaussian elimination or Gauss-Jordan elimination to find the amount borrowed at each rate. How much money was borrowed at 8%? How much money was borrowed at 10%? How much money was borrowed at 12%?

Let Z={1,2,3,4}. Let A be the following group of permutations on Z, {(1), (23),(24),(34),(234),(243)} Prove that the subgroup B={(1), (234), (243)} is normal. Give the operation table for the quotient group A/B.

Suppose v₁, v2, v3 is an orthogonal set of vectors in R5. Let w be a vector in Span(v₁, v₂, v3) such that v₁.v₁ = 18, v₂.v2=131, v3.v3=16, w.v₁ 90, w.v₂=524, w.v3=-32, then w= v₁+ V₂+ v3.

Consider the quadratic form q(x₁, x2, x3) = 6x1² + 4x1x2 +9x2² + 7x3². 3.a Find the matrix A of this quadratic form. What is the definiteness of q(x)? 3.b Find the points on the surface q(x) =1 that are closest to the origin. 3.c Find all values of k for which the surface x^T(A-kI3)x = 1 is a hyperboloid of two sheets.

Matrix A is singular. True or false? A= 1 2 -1 2 -1 3 3 1 2 TRUE FALSE

Find B-A A= 8 4 B= -7 5 9 9 4 -3 B-A=

Find a) the characteristic equation and (b) the eigenvalues and eigenvectors of A below, then decide whether or not A is diagonalizable. If not, explain why not. A= [5 8 16] [4 1 8] [-4 -4 -11]

Write an augmented matrix for the following system of equations, and state the dimension of the augmented matrix. 3x-2y+3z = -1 3x-9y+7z = 7 9y-9z = -2

The subset in the form [x y], where x and y are real numbers, is a subspace of R2 if Select one: x = 0 and y ≠ x. None of these x ∈ R and y ≤ 0. x = -y and y ≥ 0. x ≠ 0 and y ∈ R.

Solve the linear system by Gaussian elimination: x- y + 2z - w = -1 2x - y-2z- 2w = -2 -x+2y-4z + w = 1 3x + 2aw = -3 Determine for which values of a the system is consistent, and if so whether the solution is unique. Answer "inconclusive" if there is not enough information to make a decision.

Provide the missing information. Explain the meaning of the notation R₂ = R3.

The real zeros" can be described as follows (if the zero is not an integer, type the fraction form, not the decimal form A real zero is Does the graph cross the x-axis or bounce off it at this zero? Does the graph "flatten out" at this zero? Another real zero is Does the graph cross the x-axis or bounce off it at this zero? Does the graph "flatten out" at this zero? Another real zero is Does the graph cross the x-axis or bounce off it at this zero? Does the graph "flatten out" at this zero?

Construct the equation of a 4th degree polynomial with real coefficients that has the following characteristics: • the only zeros are 1/3, -4, and 2 • the graph should "bounce off" the x-axis at x = -4 • the y-intercept is (0, 32) Please, you do not have to spend time multiplying out the polynomial. Factored form is completely acceptable.

Question 1 [20 pts] Let W be the set of polynomials of the form at² + b²t + c². a) Does p(t) = t² + 4t - 4 belongs to W? b) Is W a subspace of P2?

Question2: (20 points): Let v₁ = [2 -2] and v₂ = [3 -2] be two vectors in R2. (a) [10 pts] Determine whether v = [2 -2] belongs to span {v₁, v₂} (b) [10 pts] Determine whether (v₁, v₂} span R².

Let A and B be row-equivalent matrices. (a) Show that the dimension of the column space of A is equal to the dimension of the column space of B. (b) Are the column spaces of A and B necessarily the same? Justify your answer.

Solve the system of equation by the method of your choice. If the the system has a unique solution, type in that answer as an ordered triple. If the system is inconsistent or dependent, type in "no solution". -2x - 5y + 4z = 21 -5x - 5y + z = 21 - 4y - 4z = 8.

4. The set W of polynomials of the form a₁ + a₁x + a₂x² + α3x³ + α₁x4, where a = a₂ and a₁ = a3 is a subspace of P4. Find dim(W) (you must show how you got your answer).

Algebra S = {(6,-7,8,6),(4,6,-4,1)} (a) u = (-42,113,-112,-60) check whether the set S is a basis or not

Solve the system. x+y = -4 y+z=1 x-z = -5 Is the system inconsistent, dependent or does it have 1 solution