Matrices & Determinants Questions and Answers

Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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.)
Algebra
Matrices & Determinants
Find the transformation, given by the matrix A, which stretches by a factor 5 and reflects over the line y = x.
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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:
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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?
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
→ 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.
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
(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.
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
Question 8. Given internal angle A = 30° and side b= 10in. What side length a results in a unique triangle ABC?
Algebra
Matrices & Determinants
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.)
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
Write the number in standard notation, without exponents. 7.4x10^7
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
Identify the coordinate space to which P6 is isomorphic.
Algebra
Matrices & Determinants
Let f(x) = 3x - 1 and g(x) = x² - 4. Find (fog)(1).
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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?
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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)
Algebra
Matrices & Determinants
Find a basis for the solution space for AX = 0 where A What is the dimension for this solution space?