Matrices & Determinants Questions and Answers

Use the Gauss-Jordan method to solve the following system of equations. x+y- z+ 3w= -14 3x-y+z+w = 6 4x - 2y + z- 3w= 14 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is ..., in the order x, y, z, w. B. There are infinitely many solutions. The solution is (...w), in the order x, y, z, w, where w is any real number. (Simplify your answers. Use integers or fractions for any numbers in the expressions.) C. There is no solution.

Use the echelon method to solve the given system of two equations in two unknowns. Check your answers. X/3 + y = 5/3 x/7 + y =-3/7 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution of the system is (Simplify your answer. Type an ordered pair.) B. There are infinitely many solutions. The general solution is y), where y is any real number. OC. There is no solution.

A plane traveled 4760 miles with the wind in 8.5 hours and 4420 miles against the wind in the same amount of time. Find the speed of the plane in still air and the speed of the wind.

At a concession stand, five hot dog(s) and four hamburger(s) cost $14.50; four hot dog(s) and five hamburger(s) cost $14.75. Find the cost of one hot dog and the cost of one hamburger. What is the cost of one hot dog?

Assume that the function f is a one-to-one function. (a) If f(2)= 8, find f-¹(8). Your answer is (b) If f¹(-7)= -6, find f(- 6). Your answer is

Find all subsets of the set S = {(1, 0), (0, 1), (-1,-1)} that form a basis for R2. (Select all that apply.) {(0, 1)) {(1, 0), (0, 1), (-1, -1)} {(1, 0), (0, 1)} ((-1,-1)} {(1, 0), (-1, -1)} {(0, 1), (-1, -1)} {(1, 0))

Sketch a right triangle with θ as the measure of one acute angle. Find the other five trigonometric ratios of θ. sin θ = 5/7

By further differentiation, identify lines with minimum y = 12 x² - 2x, y = x² + 4x + 1, y = 12x - 2x2, y = -3x² + 3x + 1.

Use a graphing calculator to find the values of cosine and sine of each angle. Round your answers to the nearest hundredth. a. -95⁰ b. 154⁰

A sports drink was offered for sale at $3.95 at Feng's Store. At Abreu Menezes' Store, the regular selling price of a similar sports drink was $4.35. What rate of markdown would Abreu Menezes' Store have to offer to sell the drink at the same price as Feng's Store?

Solve the following System using LU factorization x₁ + 3x₂ + x3 = 4 2x₁ + 2x₂ + x3 = -1 2x₁ + 3x₂ + x3 = 3

Solve the following System using x=A¹ b X₁ + 3x₂ + x3 = 4 2x₁ + 2x₂ + x3 = -1 2x₁ + 3x2 + x3 = 3

Find a basis for the kernel of the linear operator: T(x, y) = (3x-y, -12x + 4y) NOTE: Using the parameter t, complete the following solution.

Use a system of three equations in three variaties to solve the problem, A woman invested a $23,000 rollover IRA account in three banks paying 5%, 0%, and 7% annual interest. She invested $1.000 more at 6s than a The Smal interest she earned was 11,486. How much did she invest at each rate? amount invested at 5% $ amount invested at 6% amount invested at 7

Find transition matrix from B = {(1, 1, 1), (1, 1, 0), (1, 0, 0)} to C = {(1, 1, 1),

Recall that Rm denotes the set of all m x n matrices with real number entries. How many of the following assertions are correct? (i) In a product of matrices AB the matrix B maps rows of A to rows of AB. (ii) In a product of matrices AB rows of A are multiplied by columns of B. (iii) A matrix A of size m x n maps n-columns x to m-columns Ax.

Write the cofactor expansion for a 4 x 4 matrix down the 1st column. The terms in your expansion should look like a_{ij}C_{ij} (or similar) with specified values for i and j.

Find the dimension and a basis for the solution space. (If an answer does not exist, enter DNE for the dimension and in any cell of the vector.) x₁ + x2 - 4x3 = 7 2x13x₂ + 2x3 = 19

Consider the line parameterized by x=5-2t, y = 3+ 8t, z= 10t. Find an equation of a plane so that the plane is perpendicular to the line and through the point (4, 3, 3).

Solve the linear equation system, find its particular solution and the fundamental solution of its corresponding homogeneous linear system. Find the general solution. 2x14x2 + 5x3 + 3x4 = 1 3x16x2 + 4x3 + 2x4 = 3 4x18x2 + 17x3 + 11x4 = -1

A building that is the nearest 50.65 feet. tall has a shadow 55.48 feet long. Find the angle of the Sun to the Sen elevation of hundreth of a degree.

Show that there are no square matrices A and B such that AB-BA=1.

Find a basis for the orthogonal complement of the subspace of R4 spanned by the vectors. V₁ = (1, 4, -4, 3), V₂ = (3, 11, 3, 5), V3 = (1, 3, 11, -1) The basis for the row space is

Maximize P = x + 2y - z subject to 2x + y + z ≤ 14 4x + 2y + 3z ≤ 28 2x + 5y + 5z ≤ 30 x ≥ 0, y ≥ 0, z ≥ 0

Maximize Q = xy, where x and y are positive numbers such that x+6y² = 2.

Us a the graph of y = ex to evaluate e 2 to four decimal places.

Use Cramer's rule to solve the following system of equations. If D = 0, use another method to complete the solution. 5x + 3y = - 4 2x + 3y = -7 Write the fractions using Cramer's Rule in the form of determinants. det det det det X = y = ***

Determine whether the statement is true or false. x + 2 is a factor of P(x) = x³ + 2x² - x - 2.

Solve the system using Gaussian elimination or Gauss-Jordan elimination. x1- 5x2 + 2x3 = 6 5x2 - 2x3 = -1 -3x2 - 4x3 - 5x4 = -30 5x₁ + 4x2 + 4x3 + 5x4=56 Select one: O a. ((-5, -1,3,3)} O b. {(5, 1, -3, -3)} O c. {(5, 1,3,3)} O d. {(-5, -1, -3, -3)}

Let A = 1 -2 Q = 1/√2 -1/√2 1 4 1/√2 1/√2 We obtained the column vectors of Q by applying the Gram-Schmidt process to the column vectors of A. Find a QR-decomposition of the matrix A. NOTE: Enter exact answers. Q= R=

The matrix A= 1 -1 1 1 defines a transformation of R². Select all true statements about this transformation. A A rotates every vector by 45° anti-clockwise. A halves the area. have an inverse? A rotates every vector by 45° clockwise. A stretches every vector by the same factor. For which value or values of a does the matrix A= x -x -x x ○ x = -1,0,1 O No values of a O x = 0 O x = = 1

List five vectors in Span (V₁ V₂}. Do not make a sketch v1= 9 v2= -6 2 1 -7 0 List five vectors in SpanVv₁.₂} (Use the matrix template in the math palette. Use a comma to separate vectors as needed. Type an integer or a simplified fraction for each vector element. Type each answer only once.)

Find the transformation, given by the matrix A, which stretches by a factor 5 and reflects over the line y = x.

We start with functions y₁= 5x and y₂ = 2x²+3x. First, explain to me why the set {y₁, y2, Y3} where y3 = x²-3x is a linearly dependent set. Second, give me an example of a function y4 in the space so that {y₁, y2, Y4} is linearly independent.

be T: R3 R² a linear transformation that satisfies T(1,1,4) =(-1,-1) and T(3,-6,4)= (1,1) if v = (15,-21,4) then T(v) = (x,y) where "y" is:

For each of the systems of equations that follow, use Gaussian elimination to obtain an equivalent system whose coefficient matrix is in row echelon form. Indicate whether the system is consistent. If the system is consistent and involves no free variables, use back substitution to find the unique solution. If the system is consistent and there are free variables, transform it to reduced row echelon form and find all solutions. (a) 2x1-3x2 + 4x3 = -12 x₁-2x₂ + x3 = -5 3x1 + x2 + 2x3 = 1 (b) x1 - 5x₂ = 6 3x1 + 2x2 = 1 5x1 + 2x₂ = 1 (c) x1 + 2x2 + 3x3 + 2x4 + 15x5=1 2x1+4x2-x3 + 2x4 + 8x5 = 6 3x1+6x2-x3 + 3x4 + 13x5 = 8

a) For the following set of linear equations: 10x₁ + 25x2x3 = 2 -3x₁ - 6x₂ + 2x3 = -6 x₁ + x2 + 15x3 = -4 a) Show two iterations of the Gauss Siedel Method. Use an initial guess of x₂ = x3 = 0.5. Show your calculations in MATLAB. b) Show two iterations of the Jacobi method to solve these linear equations. Use an initial guess of x₁ = x₂ = X3 = 0.5. c) Which method produces a solution closest to the actual solution after 2 iterations?

Let A and B denote matrices whose sizes are appropriate for the following sums and products, which of the following is correct ( ) (A) (AT)T=A- (B) |A|=|AT| (C) (AB)T=ATBT (D) (AT)T=AT

Let A be a 4x4 matrix with row vectors a₁, a2, a3, a4 and determinant 2. Find the determinant of the matrix with row vectors a₁ + a₁, a₁ + a2, a₁ + a3, and a₁ + a4 in that order. 16 4 32 0

Suppose A is a 4 x 3 matrix, B is a 3 x 5 matrix, and C is a 4 x 5 matrix. Which of the following matrix operations are defined? Select ALL that apply. det(ATA) trace (BCTA) CB-T6A 2B+5C

Let A be a 6 x 6 matrix with characteristic polynomial det (λI - A) = λ²(λ+3)(λ-2)³. Which of the following statements are True? Select ALL that apply. The matrix λ is invertible. If the eigenvalue λ = 2 has geometric multiplicity equal to 3, then λ is diagonalizable. The eigenvalue λ = -3 has geometric multiplicity equal to 1. The eigenvalue λ = 2 has algebraic multiplicity equal to 3.

→ 4. Let L: R5 R¹ be a linear transformation, and let S = (v₁, 2, 3} be an indexed subset of R. Suppose that {L(v₁), L(v₂), L(vs)} is linearly independent. Show that S= {₁, 2, 3} is linearly independent.

Let S = {v1, v2, v3} be a subset of an inner product space V. Show that if S orthonormal set, then S is linearly independent.

(3 points) Let S = {₁, 2, 2 be a subset of an inner product space V. Show that if S orthonormal set, then S is linearly independent.

2. Solve each of the following initial value problems: (a) y=-y₁+ 2y₂ y/₂ = 2y₁ - y₂ yı(0) = 3, y₂(0) = 1 (b) y yi-2y2 = y₂ = 2y₁ + y₂ yı(0) = 1, y2(0)=-2 (c) y = 2y₁ - 6y3 1/₂ = y₁ - 3y3 = y2 - 2y3 yı(0) = y₂(0) = y3 (0) = 2

1. (5 points) Let P3 be the vector space of all polynomials of degree less than 4, with an orthonormal basis {f1, f2, f3, fa}. Let f € P3 be such that (f.f₁)= Given that f(1) =j², find f(1). 1 , j = 1,2,3,4.

A system of three linear equations in three variables is inconsistent. How many solutions to the system exist? O none O one O three O infinitely many

Several customers order small fruit baskets filled with apples, bananas, and oranges. Let a represent the price per pound of apples, b represent the price per pound of bananas, and c represent the price per pound of oranges. The system represents the number of pounds of each type of fruit and the total price of each fruit basket. How much per pound does each type of fruit cost? O apples: $2.00/lb, bananas: $0.50/lb, and oranges: $3.00 O apples: $2.00, bananas: $1.50/lb, and oranges: $3.00/lb O apples: $2.50/lb, bananas $0.25/lb, and oranges: $3.00/lb O apples: $2.50, bananas: $0.75, oranges: 3.00/lb

Which classification describes the following system of equations? inconsistent and dependent O consistent and dependent consistent and independent O inconsistent and independent

Which system is independent and inconsistent? [x-y+z=2 O x-y-z = 2 x+y+z=2 x-y+z=2 Ox+y-z=3 |x-y-z=4 O O [2x+2y+2z=4 -x-y-z=-2 x+y+z=2 x-y-z=2 x+y-z=3 |-x+y+z=4 M tal