Matrices & Determinants Questions and Answers

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]

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.

The corresponding linear combination would be: is a linear combination of the set S containing the vectors

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

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.

Given v₁ and v₂ in a vector space V, let H= Span {v₁, v₂}. Show that H is a subspace of V.

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.

Write the system of linear equations in the form Ax= b and solve this matrix equation for x. -2x₁-3x₂ = -11 6x₁+ X₂=-39

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

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?

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: (-∞, ∞)

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?

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.

If u and v are vectors in R³, then ||u – v|| = ||u|| - ||v||. (a) True (b) False

Find the vector equation of line of intersection of the planes 2x+2y+2z=4 and 2x-y+3z=1.

Find the inverse: f(x) = ln(7x + 3). Present the answer using function notation.

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

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).

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:

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.

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

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.

(T/F) A matrix A is invertible if and only if 0 is an eigenvalue of A.

Is the subset below independent? Support your answer. {(1, 1, 1, 1), (2, 0, 1, 0), (0, 2, 1, 2)} in R4

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?

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

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.

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:

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)

Write a matrix for rotation of the plane R² clockwise by 60°. Show that the matrix is orthogonal.

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.

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.

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.

x+y+z= 14 For the system x -z = 5, what is the corresponding augmented matrix 2x - y = 3 before solving?

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.

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

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}?

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.

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

Let A be an invertible 2 x 2 matrix. Find a general formula for A-¹ in terms of the entries of A.

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.)

Rewrite log ₃ 64 as a logarithmic expression with an argument of 4.

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])

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.

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=

Find all unit vectors u E R³ that are orthogonal to both v₁ = (2,7,9) and V₂ = (-7,8,1).

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.

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.

Rewrite log 3 √10 as a a logarithmic expression with an argument of 10.

Find the equation y = Bo + B₁x of the least-squares line that best fits the given data points. (4.3). (5,2). (7.1), (8,0)