Matrices & Determinants Questions and Answers

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.

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 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

Question 8. Given internal angle A = 30° and side b= 10in. What side length a results in a unique triangle ABC?

Given a planar graph G such that all the vertices have even degrees: Prove that all the regions separated out by this graph (all the faces) can be 2-colored. That is, the faces can be colored such that no two faces which share an edge can have same color. (Hint: Use induction. Look at outer rim of the graph and think about removing all the edges of the outer rim.)

The manufacturer of a wine bottle spends $5.20 to make each bottle and sells them for $10.40. The manufacturer also has fixed costs each month of $5000. Find the cost function, C, when a bottles are manufactured Find the revenue function, R, when a bottles are sold

Define T: P₂ P₂ by T(ag + a₁x + a₂x²) = (-3a₁ + 5a₂) + (-4a0 +4a₁-10a2)x+ 5a₂x². Find the eigenvalues. (Enter your answers from smallest to largest.) (A1, A2, A3) -2,4,6 Find the corresponding coordinate eigenvectors of T relative to the standard basis (1, x, x²). IT

Write the number in standard notation, without exponents. 7.4x10^7

Find a basis for the orthogonal complement of the subspace of R4 spanned by the vectors. V₁ = (1, 5, -4,4), v₂=(2, 9, 0, 2), v3 = (1,4,4,-2) The basis for the row space is

Identify the coordinate space to which P6 is isomorphic.

Let f(x) = 3x - 1 and g(x) = x² - 4. Find (fog)(1).

Determine two vectors that are perpendicular to each other and also perpendicular to u= [2, 7, -1]. It is also known, that one of these vectors, creates two equal angles with x and y axes, and the rectangular prism formed by these three vectors has the volume 4482 m³.

Sean, Angelina, and Sharon went to an office supply store. Sean bought 7 pencils, 8 markers, and 7 erasers, His total was $22.00. Angelina spent $19.50 buying 4 pencils, 8 markers, and 6 erasers. Sharon bought 6 pencils, 4 markers, and 7 erasers for $17.75. What is the cost of each item?

Use Gaussian elimination to find the values of a for which the equation system x + y = 2z = a +7 3x - y + az = -3 -x + ay - 4z = 8 has (i) exactly solution; (ii) more than one solution; (iii) no solution.

Find the amount of time required for a $ 21,000 investment to double if the annual interest rate r is 9.2% and interest is compounded continuously. Round your answer to the nearest hundredth of a year. A 115.71 years B 1.08 years C 7.53 years D 108.18 years

Find the determinant of the following matrix using the co-factor expansion method. A= 1 -1 2 -2 2 -3 1 1 -3 0 0 0 -2 -1 1 2 Note: You may use the cross-multiplication method to calculate the 3x3 minor, i.e. the determinant of the 3x3 matrix.

When using elimination and substitution, ex-plain how to recognize a system of linearequations that has no solutions. When using elimination and substitution, ex-plain how to recognize a system of linearequations that has infinitely many solutions.

1 -1 2 -3) Given the following matrices A = 3 B = (²-1). 2 [i] Calculate 2A - B. [ii] Find AT [iii] Calculate AB. [iv] Based on your results from part [iii], what can you conclude about the relationship between matrices A and B. and

The cost for three packages of moving boxes is modeled by the system of equations below. Let s represents the cost of each small box, m represents the cost of each medium box, and / represents the cost of each large box. Which ordered triple (s, m, ) represents the costs of the three boxes? 7s+4m+21=24 5s+3m+6/= 30 3s+7m+ 10/=46 O (1.00, 2.50, 3.50) O (1.00, 2.00, 2.90) O (1.50, 2.00, 2.75) O (1.50, 2.00, 2.00)

Find a basis for the solution space for AX = 0 where A What is the dimension for this solution space?