# Matrices & Determinants Questions and Answers

Algebra
Matrices & Determinants
Write "T" or "F" to indicate true or false if vector u is a linear combination of the vectors in the set S. If υ can be expressed as a linear combination, write out the corresponding linear combination; otherwise, place 0 in all corresponding boxes. _________: υ=[11/-3/5/20] is a linear combination of the set S containing the vectors [2/-5/4/-3] [-5/3/-5/-5] [4/-5/4/2] The corresponding linear combination would be: [11/-3/5/20]=______[2/-5/4/-3]+ [-5/3/-5/-5]+ [4/-5/4/2]
Algebra
Matrices & Determinants
Determine whether the eigenvalues of each matrix are distinct real, repeated real, or complex. [7/-20 +4/-11] [3/3 -4/1] [26/-60 +12/-28] [-1/-4 +/1-5]
Algebra
Matrices & Determinants
Suppose matrix A is a 4 x 4 matrix such that A. [-18/24/36/-24]=[-3/4/6/-4] Find an eigenvalue of A.
Algebra
Matrices & Determinants
The corresponding linear combination would be: is a linear combination of the set S containing the vectors
Algebra
Matrices & Determinants
Let T: R³→ R³ be a linear transformation defined by T(v) = Av, 1 0 0 A = 1 0 -1 Find T(3,1,-1) 2 1 -2
Algebra
Matrices & Determinants
4 1 2 A = 0-3 3 0 0 2 Describe the set of all solutions to the homogeneous system Ax = 0. Find A⁻¹, if it exists. a) Describe the set of all solutions to the homogenous system Ax = 0 b) Find A⁻¹, if it exists.
Algebra
Matrices & Determinants
Given v₁ and v₂ in a vector space V, let H= Span {v₁, v₂}. Show that H is a subspace of V.
Algebra
Matrices & Determinants
Solve the following system analytically. If the equations are dependent, write the solutions set in terms of the variable z. x-y+z= -7 8x+y+z=8 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is one solution. The solution set is {}. (Type an integer or a simplified fraction.) B. There are infinitely many solutions. The solution set is {(z)}, where z is any real number. (Simplify your answer. Use integers or fractions for any numbers in the expressions.) C. The solution set is Ø.
Algebra
Matrices & Determinants
Write the system of linear equations in the form Ax= b and solve this matrix equation for x. -2x₁-3x₂ = -11 6x₁+ X₂=-39
Algebra
Matrices & Determinants
Suppose that ѵ₁ = (2,1,0,3), ѵ₂ = (3,-1,5,2), and v₃= (-1,0,2,1). Find the vector spanned by vectors ѵ₁, ѵ₂ and ѵ₃
Algebra
Matrices & Determinants
The probability of event A is 0.43, and the probability of event B is 0.55. What is the probability of both occurring if A and B are independent events?
Algebra
Matrices & Determinants
Determine the domain and range of the function f(x) = ln (-x+5). Select the correct answer below: a) Domain: (-∞, -5); Range: (-∞,∞) b) Domain: (5,∞); Range: (-∞,3) c) Domain: (-∞, 5); Range: (-∞,∞) d) Domain: (-∞, ∞); Range: (-∞, ∞)
Algebra
Matrices & Determinants
Consider the set X= (2,3) U (8,10) 1. show that this set is open. 2. show that this set is not convex. 3. consider set Y=[3,8]. Is Y an open set? is X U Y open? Why/ Why not? 4. Is X and Y connected? WHY/WHY NOT?
Algebra
Matrices & Determinants
For given equations 10x₁ + 2x₂-x₃ = 27 - 3x₁ - 5x₂ + 2x₃ = -61.5 x₁ + x₂ + 6x₃ = -21.5 (a) Solve by naive Gauss elimination. (b) Solve by LU factorization. (c) Solve by iterative method.
Algebra
Matrices & Determinants
If u and v are vectors in R³, then ||u – v|| = ||u|| - ||v||. (a) True (b) False
Algebra
Matrices & Determinants
Find the vector equation of line of intersection of the planes 2x+2y+2z=4 and 2x-y+3z=1.
Algebra
Matrices & Determinants
Find the inverse: f(x) = ln(7x + 3). Present the answer using function notation.
Algebra
Matrices & Determinants
cos θ=(2/3), tan θ <0 Find the exact value of sin θ A. - √5 B. -√5/2 C. -√ 5/3 D. -3/2
Algebra
Matrices & Determinants
Let f be: R²→R given by f(x,y) = (y - x²) (y – 2x²) Show that f has no local minimum at (0,0), but that every restriction flg for every line G⊂R² through the origin, has a local minimum at (0,0).
Algebra
Matrices & Determinants
Tina invested \$8, 550 in a 4-year CD that earns 3% annual interest that is compounded continuously. How much will the CD be worth at the end of the 4-year term? Include a dollar sign in your answer and commas when appropriate. Round to the nearest cent. Provide your answer below:
Algebra
Matrices & Determinants
Let v = [2, 0, -1] and w = [0, 2, 3]. Write w as the sum of a vector u₁ parallel to v and a vector u₂ orthogonal to v.
Algebra
Matrices & Determinants
Kindly help with the Answer to the question mentioned below. Thank you. If G is a group of order 208 then show that there exists a normal subgroup of order 27 or 9
Algebra
Matrices & Determinants
Let W be the subspace of R4 generated by the vectors (1, -2, 5, -3), (2, 3, 1, -4), and (3, 8, -3, -5). Find a basis and the dimension of W.
Algebra
Matrices & Determinants
(T/F) A matrix A is invertible if and only if 0 is an eigenvalue of A.
Algebra
Matrices & Determinants
Is the subset below independent? Support your answer. {(1, 1, 1, 1), (2, 0, 1, 0), (0, 2, 1, 2)} in R4
Algebra
Matrices & Determinants
At the county 4th of July fair, a local strong man challenges contestants two at a time to a tug-of-war contest. Contestant A can tug with a force of 390 pounds. Contestant B can tug with a force of 320 pounds. The angle between the ropes of the two contestants is 30°. With how much force must the local strong man tug so that the rope does not move?
Algebra
Matrices & Determinants
Solve the system of linear equations. (If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter your answer in terms of a.) x+4z = 9 4x - 2y + z = 11 2x - 2y - 7z = -7
Algebra
Matrices & Determinants
Which of the following describes H = Span{v₁, v2, v3), where v1 =1 v2= -2 v3= 1 1 -1 4 -3 7 0 A. H is a plane through the origin. B. H = R³. C. H is a line through the origin. D. H is a line not through the origin. E. H is a plane not through the origin.
Algebra
Matrices & Determinants
Let f: R² → R² be the rotation transformation about the origin by 60° counterclockwise. Let g: R² → R²³ be defined by g(x, y) = ( −x/2+my,mx+y/2) 1. Find m ER so that fo g is the reflection in the y- m= 2. With this m, find the image of the point A(-4, 5) through go f. 3. Find the dimension of ker(go fogof). Answer:
Algebra
Matrices & Determinants
Find the equation of the hyperbola centered at (-1,4), vertices at (-1,-3) and (-1,11) and foci at (-1-5) and (-1, 13)
Algebra
Matrices & Determinants
Write a matrix for rotation of the plane R² clockwise by 60°. Show that the matrix is orthogonal.
Algebra
Matrices & Determinants
5. Consider the following system of linear equations: x₁ - x₃ - 2x₄ - 8x₅ = -3 -2x₁ + x₃ + 2x₄ + 9x₅ = 5 3x₁ -2x₃ -3x₄ -15x₅ = -9 a) Write the system in the matrix-vector form by finding the coefficient matrix A, variable vector x, and the right-hand-side vector b. Also form the augmented matrix for the system.
Algebra
Matrices & Determinants
Consider the matrix equation Ax = b where, a b A= 1 1 where a, b>0. (a) When is A strictly diagonally dominant and what does this tell us about the Gauss-Seidel method for Ax = b. (b) Find the iteration matrix for the Gauss-Seidel method applied to Ax = b.
Algebra
Matrices & Determinants
Compute the determinant of the n x n matrix A = (aj) such that, for all i, j, aij = 3 if i j and aij = 2 if i ≠ j.
Algebra
Matrices & Determinants
x+y+z= 14 For the system x -z = 5, what is the corresponding augmented matrix 2x - y = 3 before solving?
Algebra
Matrices & Determinants
Find a subset of the vectors v₁ = (0,2,2,4), v₂ = (1, 0,-1,-3), v₃ = (2,3, 1, 1) and v₄ = (-2, 1, 3, 2) that forms a basis for the space spanned by these vectors. Explain clearly.
Algebra
Matrices & Determinants
Consider the following matrix: A = 4 0 1 -1 1 0 -2 0 1 a) Find the eigenvalues of A b) Using the eigenvalues of A, find the corresponding eigenvectors
Algebra
Matrices & Determinants
Let v₁= [2 v₂= [10 v₃= [-6 , and y= [-4 -1 -4 1 3 -1] -7/2] 0] h/2] For what value(s) of h is y e span {v₁,v₂,v₃}?
Algebra
Matrices & Determinants
Determine whether the set {Pᵢ, P₂, P₃} is linearly independent in P₂, where p₁ = 2+x+3x² + 4x³, p₂ = 4 + 3x + 2x² + x³ and p₃ = 1 + 2x + 3x² + 4x³. Show all working.
Algebra
Matrices & Determinants
Write the system of linear equations in the form Ax=b and solve this matrix equation for x. x₁-5x₂+2x₃=-5 -3x₁+x₂-x₃=2 -2x₂+5x₃=11
Algebra
Matrices & Determinants
Let A be an invertible 2 x 2 matrix. Find a general formula for A-¹ in terms of the entries of A.
Algebra
Matrices & Determinants
Let the function f be given by f(x) = x³ − 5x² + 9x − 4 and A = 1 -1 0 2 3 -1 -1 0 1 Show that A satisfies f(x) = 0. (HINT: Show that A³ -5A² +9A-4I₃=0.)
Algebra
Matrices & Determinants
Rewrite log ₃ 64 as a logarithmic expression with an argument of 4.
Algebra
Matrices & Determinants
2) Shou that the Projection onto the vector v = [1, -2,1] is a linear transformation T: R³ R3 b) Find the Standard matrix [T] for this transformation C) Find the nullity ([T]) and rank ([T])
Algebra
Matrices & Determinants
Let Q be an nxn invertible matrix and {u₁, U2,..., uk} is a set of vectors in R. Prove that {u₁, 12,..., uk} is linearly independent if and only if {Qui, Qu₂,..., Que} is linearly independent.
Algebra
Matrices & Determinants
Find a formula in terms of k for the entries of Ak, where A is the diagonalizable matrix below. A= -3 -4 2 3 A^k=
Algebra
Matrices & Determinants
Find all unit vectors u E R³ that are orthogonal to both v₁ = (2,7,9) and V₂ = (-7,8,1).
Algebra
Matrices & Determinants
Consider a scalar valued function g(W) = 4w₁₁ +6w₁₂+6w₂₁+9w₂₂ of the 2×2 matrix W = {wij}. (1) Express the matrix form of g (W). (2) Find the matrix gradient of g (W) with respect to W.
Algebra
Matrices & Determinants
Suppose the augmented coefficient matrix of a certain linear system is: [1 1 3 Ⅰ -3] [1 2 -2 Ⅰ 1] [3 9 k Ⅰ 16] for some k∈ R. For what value(s) of k will the system have no solution? k =? For what value(s) of k will the system have infinitely many solutions? k= =? Enter your answers as comma-separated lists. If there are no values of k satisfying the given condition, enter NONE.
Algebra
Matrices & Determinants
Rewrite log 3 √10 as a a logarithmic expression with an argument of 10.