Matrices & Determinants Questions and Answers

If u and are linearly independent, then u +, u are also linearly independent. True False Justification:

A doctor's prescription calls for a daily intake containing 40 mg of vitamin C and 10 mg of vitamin D. Your pharmacy stocks two compounds that can be used: one contains 60% vitamin C and 10% vitamin D, the other 40% vitamin C and 20% vitamin D. How many milligrams of each compound should be mixed to fill the prescription? To fill the prescription, mix _ mg of the first compound with _ mg of the second compound.

Consider the following 2*2 system: 2x₁ - x₂ = 1 -x₂ + 2x₂ = 0 a) Rewrite the linear system in the matrix format Ax=b . b) Solve Ax=b with inner product, and show the row picture for this system. c) Solve Ax=b with linear combination of columns in matrix A, and show the column picture for this system.

why is a system with 50 equations and 51 variables is guaranteed to contain atleast one varable?

Let a= (1, 2, 5) and b = (3, 3, k). Find k so that a and b will be orthogonal.

(a) The zero vector is orthogonal to all subspaces. (b) If one of the eigenvalues is zero, the matrix is singular. (c) Suppose the eigenvalues are 1, 2, 3, the matrix is NOT invertible. (d) Cramer's rule can be applied to solve Ax=b when A is singular. (e) If two eigenvalues are the same, the matrix is NOT be diagonalizable. true or false?

Consider the set W of all 2 x 2 matrices A such that both (1, 2) and (2, -1) are eigenvectors of A. Prove that W is a subspace of the space of all 2 x 2 matrices and find the dimension of W.

A real quadratic form in three variables is equivalent to the diagonal form 6y + 3y+ Oy. Then the quadratic form is Positive semi-definite Positive definite Indefinite Negative definite

Factor completely, or state that the polynomial is prime. x² +81 Select the correct choice below and fill in any answer boxes within your choice. A. x² +81 = B. The polynomial is prime.

Solve the formula for the variable h. f=9gh h=

Construct a triangle of sides 68, 58 and 42 mm long respectively. Bisect the angles to find the centre of a circle that will touch the three sides of the triangle. Draw the circle and state its radius.

Take A= 3, B=1, C=4 Find the orthogonal projection matrix P onto the plane ax + by - cz = 0 (a, b, c are defined in the instructions)

1 3 0 0 2 4 1 5 4 1 1-1 a) (7 points) What is the basis for the Row(B)? 3. Let B= b) (7 points) What is the basis for the left null space of B

Find an orthogonal basis for the column space of the matrix to the right. -1 5 5 1 -7 4 1 -1 7 1 -3 -4 An orthogonal basis for the column space of the given matrix is (Type a vector or list of vectors. Use a comma to separate vectors as needed.)

Solve the initial va problem xi-Axt) for 120, with x0) (2.1) Classify the nature of the origin as an attractor, repeller, or saddin point of the dynamical system desobed by Find the actions of greatest attraction and/or repulsion Solve the initial value problem Classify the nature of the origin as an attractor, repeller, or saddle point Choose the correct answer below

Suppose that S={(-1,1,0), (2,2,0), (0,0,3)} is orthogonal basis for a subspace W of R³. Find the orthogonal projection of u in W where linear combination of the basis vector is u =(2,1,3).

Consider the two points A = (-1,2) and B = (1,8) to be points on the curve. a) Give a possible formula for the function of the form y = a(b) that passes through these two points. b) Find the domain for the function you have found in part a) c) Find the asymptote for the function you found in part a) d) Find the x- and y-intercepts if any. e) Graph the function you have found in part a).

Using the matrix U = [U₁U₂U3], determine whether or not the columns of U span R³. Explain your approach. Solution: %code

Write the decomposition of a vector X(-5; 4; 0), (0; 1; 2), (1; 0; 1), F(-1; 1; 0). Solve the task and fill in the required fields. Ansver: A= A₁= X₂= in terms of vectors P. 9,7 if: X =0

Write the decomposition of a vector X(-5; 4; 0). p(0; 1; 2). ā (1; 0; 1), 7 (-1; 1; 0). Solve the task and fill in the required fields. Answer: 4₁= x₁= in terms of vectors P. 9, F if: X =(

What is the solution of the system of linear equations [a₂x+₂y=b₂ [a₂₁²x + a₂₂y = 6₂ a. The solution of SLE is a set of the ordered pairs of numbers (y; x). b. The solution of SLE is a set of the ordered pairs of numbers (x, y).

Solve the following system {3x + 5y - 7z = 8 (mod 83) 9y + 13z = 13 (mod 83) 8x-9Y+13z=13 (mod83) 7x + 4y + 5z = 15 (mod 83) }

Simplify the expression by combining like terms. 7m+4m

Let V=R² and B be the ordered basis, B =[ 1 [-1 1 ] 1] Find the coordinates of the vector, V = 1 5 relative to B.

Select the inequality that is equivalent to the given inequality. -7x ≤14 Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer. a x ≥ -2 b x ≥ 2 c x ≤ -2 d x ≤ 2

Solve the equation using the quadratic formula. 2x²-9x-1=0

We have a matrix a find a, b, c such that the matrix is a quadratic form

Determine the values of k such that the system of linear equations does not have a unique solution. (Enter your answers as a comma-separated list.) x + y + kz = 6 x + ky + z = 4 kx + y + z = 3

Use Cramer's rule to solve the system. X +y-z = -4 x - y + 2z = 14 x + 4y 2z = -17 Find the determinants. D = Dx= Dy= Dz= Select the correct choice below and fill in any answer boxes within your choice. A. The solution set is {} (Simplify your answers.) B. The system contains dependent equations. C. The system is inconsistent.

Describe how to transform the parent function y = x² to the graph of the function below. Choose all that apply. y = -2(x + 3)² + 1 Select one or more: a. reflected across the x-axis b. compressed by a factor of 2 c. translated 3 units up d. translated 3 units right e. translated 1 unit up f. translated 1 unit right

Consider the following observation model where [by 1 b₂ = 5 b3 = 3 b4 =4)= 4) (a) x2.04, 4.52,-0.24] (b) x-2.04 (c) x 6.45 (d) x [6.45, 4.52. -0.85] b₁ = I₁ +23+7²₁ b₂r2-2r3+ n₂ b3=11-423+ng b₁2r1 +223 +14 What What is the least square estimate of x= [1,2,3]?

This set of questions pertains to reference angles. a. Name 3 different angles which could have a reference angle of 20 degrees. How did you arrive at this answer? 2-3 sentences. b. What are the characteristics of a reference angle? 2-3 sentences. c. How is it possible that angles can have different measurements, but still have the exact same reference angle? 2-3 sentences.

Find a basis for, and thus the dimension of, the subspace of R³ given by the plane * - 3y + 5z = 0. Why would it make no sense to ask for a basis or dimension for the plane 3y + 5z = 1?

Evaluate (without use of a calculator). (a) log4 2 + log4 128 = (b) log4 112- log4 7

(a) Let A be an invertible n x n and suppose that A-1 = A. Prove that det(A) = -1 or det(A) = 1. (b) Let A be a square matrix such that A² -7A+8I = 0 [where I and 0 are the identity matrix and the zero matrix, respectively). Show that det(A) # 0.

(a) Suppose that A is a 3 x 3 diagonalizable matrix with a basis of eigenvectors of R³ given by {u₁, U₂, U3} with corresponding eigenvalues A1, A2, A3. Suppose that the matrix B is also a 3 x 3 diagonalizable matrix with the same eigenvectors although with possibly different eigenvalues 11, 12, 13. Prove that A + B is also a diagonalizable matrix. (b) Suppose that the square matrices A and B are similar. Prove that the matrices A2+ A+ I and B2 +B+ I are also similar. (c) Suppose that the square matrices A and B both have their characteristic polynomial equal to -³ + X. Prove that A is similar to B.

0.0116 0.0046 0.0023 0.0023 0.0046 0.0116 Let D= 0.0046 0.0093 0.0046 be a flexibility matrix, with flexibility measured in inches per pound. Suppose that forces of 40, 10, and 20 lb are applied to a beam at points 1, 0.0023 0.0046 0.0116 and 3, respectively. Find the corresponding deflections y₁, y2, and y3. #1 #2

Calculate the inverse A^-1 of the matrix A = ( -1 2 2) (2 -1 2) (2 2 -1)

Find the matrix P that orthogonally diagonalizes A. Compute P-¹AP. A = 3 2 4 2 0 2 4 2 3

Find 2A + 3B. a=[4 8] b=[-5 9] [1 7] [5 -8]

Matrix A represents the number of points scored in each quarter for the first 4 games of football played by WCHS. Matrix B represents the number of points scored in each quarter for the first 4 games of football played by WJHS. Matrix A Matrix B Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Game1 6 0 13 3 Game1 0 3 9 0 Game2 21 18 0 7 Game2 7 14 7 6 Game3 14 28 6 0 Game3 3 9 12 17 Game4 0 0 35 17 Game4 23 0 9 7 (a) What does the number 28 represent? Be specific. (b) Suppose WCHS was playing WJHS in game 3. Who won? (C) What is the AVERAGE point scored by WJHS in the first quarter?

Solve each system of equations using elementary operations. a. (i) x + y + z = 0 (ii) x - y = 1 (iii) y - z = -5 d.(i) x/3+y/4+z/5=14 (ii) x/4+y/5+z/3=-21 (iii)x/5+y/3+z/4=7

For each, determine the amplitude, the axis of the curve, the range, the period, the phase shift and the zero(s) within one period. Keep your solutions neat and organized. (7 marks each) a. y 4 sin(18x) =- 5 = b. y = -6 sin(x-40") - 3 c. y=-25 cos(60a) - 21

A n x n Hermitian matrix contains exactly k real entries. Show that k + n must be even.

Solve using Cramer's Rule. - 5x + 2y + 3z = 27 6x + 6y + 5z = - 32 -5x + 3y + 3z = 25 The solution of the system is (Type integers or simplified fractions.)

Solve using Cramer's Rule. 3x-y = 7 x-3y=9 What is the solution of the system? (Simplify your answer. Type an ordered pair, using integers or fractions.)

Solve the following system of equations using Gaussian or Gauss-Jordan elimination. - 10x-3y = - 17 4x + y = 7 Select the correct choice below and fill in any answer boxes in your choice. (A). The solution to the system of equations is (Simplify your answer. Type an ordered pair.) (B).There are infinitely many solutions. (C). There is no solution.

Without graphing, classify the system as independent, dependent, or inconsistent. 12y - 3x = 21 8y = 2x 14

rite the equations for f(x) and g(x). Then identify the reflection that transforms the graph of f(x) to the graph of g(x).

Write an exponential function to model the following situation. A population of 120,000 grows 5% per year for 14 years. How much will the popluation be after 14 years? Write an exponential function in terms of x. y=*