Matrices & Determinants Questions and Answers

Let T: R² R2 be the linear (matrix) transformation defined by: T(x1, x2) = (-7x1 + x2, 6x1-4x2,8x1 +5x2,-9x1- 2x2). Determine the standard matrix A of the linear transformation.

Let T: R² → R² be the linear (matrix) transformation defined by: T(x1,x2)=(-7x1 + x2, 6x1-4x2, 8x1 + 5x2, -9x1 - 2x2). Determine the standard matrix A of the linear transformation.

Let A = 2 9 -2 6 1 6 -1 4 Find an orthonormal basis of the kernel of A.

You are holding a kite string in your hand. The angle of elevation from your hand to the kite is 53°and the distance to the kite is 300 feet. Your hand is 5 feet above the ground. How high is the kite? Round your answer to the nearest tenth of a foot.

Suppose a firm produces three outputs, y1, y2 and y3, with three inputs z1,z2 and z3. The input-requirement matrix is given by A = [ (3 12 )(2 5 1)( 1 1 3) ] If the firm produces 10 units of y1, 20 units of y2 and 10 units of y3, how much of z1,z2 and z3will require?

For the following system of homogenous linear equations: x-y-11z=0 2y-13z = 0 Use elementary row operations to solve the above system. Identify the free variable(s) if any.

Given the planes P₁: 2xy + 5z = 8 and P₂: x-y + 2z = 4, and given the point A(5, -3,2). (a) Find the cosine of the angle between the planes P₁ and P₂. (b) Find the point on the plane P₁ closest to A. (c) Find an equation for the line containing point A that is parallel to both P₁ and P₂.

Let T: R6 → R6 be linear transformation such that T(x)= Ax and det(A) 0. Find the rank(T), range(T), kernel(T) and nullity (T) Rank (T) Range(T) Kernel(T) Nullity(T)

You are given the following set of data points. x 0 2 3 y 2 4 6 (a) We first assume that all points lie exactly on the line y = a + bx. In form of matrices, we obtain Am = y, where m =a and y=2 b 4 6 Enter the matrix A in the box below. (b) Find the least square line y = a + bx y = Enter the intercept and gradient in exact form as either an integer or fraction.

x - 2y = 23/2 x + 3y = 33/2 Solve Using a Matrix with Cramer's Rule

9. Let V be the vector space of all functions f: R R. (Note that there are no continuity assumptions on f.) Let T: VV: f(x) →xf(x). (a) Show that I is a linear transformation. (b) Show that every real number is an eigenvalue of T.

Problem 6. (1 point) If the determinant of a 5 × 5 matrix A is det(A) = 5, and the matrix B is obtained from A by multiplying the second column by 2, then det (B) =

4. Find the equation of an ellipse passing through (-4,0) with foci at (0,0)and (-1,4).

Solve the system of equations by converting to a matrix equation and using the inverse of the coefficient matrix, as in Example 6. 2x + 4y + z = 4 -x+y-z = 0 x + 4y =-6

Let R* be equipped with standard inner product. Find all vectors in R* orthogonal to v = (1, -5,8,2)

Algebra Let R* be equipped with standard inner product. Find all vectors in R* orthogonal to v = (1,-5,8,2) use augmented matrices if possible (write neatly and will upvote thank you) don't have to use matrices only if possible.

Problem 1. 15pts. Determine whether the following statements are true or false. If the statement is true, write T in the box provided under the statement. If the statement is false, write F in the box provided under the statement. Do not write "true" or "false". (a) If A and B are symmetric n x n matrices, then ABBA must be symmetric as well. If A is an invertible matrix such that A-¹ = A, then A must be orthogonal. If V is a subspace of R" and is a vector in R", then the inequality (proj) 2 0 must hold. (d) (b) (e) - If matrix B is obtained by swapping two rows of an n x n matrix A, then the equation det (B) = -det (4) must hold. There exist real invertible 3 x 3 matrices A and S such that ST AS = -A.

Find the domain and codomain of the linear transformation of the problem below. T( a b c d) = 1+(a+b)x + (b-c)x^2 + (d-c)x^3

Does set S span a new vector and is set S a basis or not? S = {(2,-1, 3), (5, 0,4)} (a) u = (1, 1, -1) (c) w = (1, -8, 12) (b) v = (8,-1,27) (d) z = (-1, -2, 2)

The temperature from midnight to noon on March 15 can be modeled using the function T(h) = 4.2(h) + 42, where T = temperature and h = hours relative to noon (note: h= 0 represents noon). Find the domain of the function. 0<h <12 -12<h <0 -12 <h < 12 -6<h <0

What is the scale factor if a 8" by 16" photograph is enlarged to a poster that is 2 ft. by 4 ft.?

Find the dimensions of the following vector spaces. (a) The vector space of all diagonal 3 x 3 matrices (b) The vector space R6 (c) The vector space of all upper triangular 2 x 2 matrices (d) The vector space P₁[x] of polynomials with degree less than 4 (e) The vector space R² (f) The vector space of 3 x 3 matrices with trace 0

Consider the following differential equation to be solved by the method of undetermined coefficients. y" - 14y' + 49y = 28x + 5 Find the complementary function for the differential equation. Yc(x) = Find the particular solution for the differential equation. Yp(x) Find the general solution for the differential equation. y(x) =

Determine whether S = {1-t,2t+3t2,t²-2t³,2+t³) is a basis for P₁.

Sean kicks a soccer ball with an initial velocity of 70 ft/sec at an angle of 41" with the horizontal. How long is the ball in the air? How far does the ball travel horizontally?

Let B=(-x+x2, -2x +x²,2-3x+x²). (a) Show that B spans P2. (b) Hence, explain why B forms a basis for P₂. (c) Find the coordinate vector of 22 relative to the basis B.

Find the matrix for the linear transformation that reflects about y = 2x. Please give a detailed explanation of how you got the map to reflect.

Establish the identity. cscθ+cotθcotθ-cscθ+cotθcscθ=sinθtanθ Question content area bottom Part 1 Subtract the fractions on the left side. Apply the appropriate Pythagorean identity to simplify the numerator. enter your response herecscθcotθ (Simplify your answer.)

Determine a basis for the subspace S of R4 consisting of all solutions to the linear system -2X1 + 3X2 - 4X3 + X4 = 0 2X1-4X2+6X3 + 2X4 = 0 What is the dimension of S?

Use the simplex method to solve. Minimize: The minimum value w = w=64y₁ +20y2 +41y3 subject to: 8y₁ + 2y2 +9y3 ≤ 15, 8y₁ +4y2 + 4y3 211 Y₁ > 0, y2> 0, y3 > 0

For each graph in #s 8 & 9, find A, B, period, C, D, and a positive sine and positive cosine equation each x unit = π/12, each y = 1 unit

The world population was approximately 7.405 billion in 2017 with an annual growth rate of 1.06%- At this growth rate, the world population P(t) (in billions) can be represented by P(t) = 7.405(1.0106), where t is the number of years after 2017. Round values to the nearest tenth when necessary. (Source: CIA World Factbook) a. P(0) = b. P(3)

Find the following for the function f(x)=2x+3/7x-8

For each of the following matrices, write down the quadratic form and decide, with proof, whether the matrix is positive definite: (a) A= 7 5 5 5 (b) B= 4 2 -1 2 0 2 -1 2 3

Let V be the vector space R³. (a) Let W = {(x, y, z) = R³ : z = x + y}. Is W a subspace of V? Give reasons. (b) Let U = {(x, y, z) = R³ : z = x²} Is U a subspace of V? Give reasons.

Determine the integers n for which the pair of congruences 2x-y = 1 and 4x + 3y = 2 modulo n has a solution.

Consider the linear map F: R¹ R³ given by F(x, y, z, w) = (x+y+z, x+y+w, 2x + 2y). 1. Find the matrix associated with F. 2. What is the dimension of the kernel of F?

Let A = A is a 4X4 matrix . Show that A is invertible by Gershgorin Circles.

Let A = A is a 4X4 matrix. Show that A is invertible by Gershgorin Circles.

Let A be a symmetric 4 x 4 matrix such that det(A) = 3. Find adj(3A)A^T.

Let B be a symmetric 5 x 5 matrix such that det(B) = 2. Find adj(2B) BT

Let A be a symmetric 4 x 4 matrix such that det (A) = 3. Find adj (3A) AT.

Use Gauss-Jordan elimination to solve the following system, 3x +9y+ 2z + 12w= 1 x + 3y - 2z + 4w= 2. -2x - 6y - 10w= 0,

Decide if the following statements are TRUE or FALSE. Write a proof for the true ones and provide a counter-example for the rest. Every linear operator T: Rⁿ → Rⁿ can be written as T = D + N, where D is diagonalizable, N is nilpotent and DN = ND.

What can be said about the minimal polynomials of AB and BA. (Hint: in the singular case consider tm(t) where m(t) is the minimal polynomial of, say, AB.)

Show that every matrix "A" with real entries can be same as symmetric and skew symmetric matrix

If A and B are invertible, then AB and BA are similar. (a) True (b) False

An investor is considering three types of investments: a high-risk venture into oil leases with a potential return of 15%, a medium-risk investment in bonds with a 9% return, and a relatively safe stock investment with a 5% return. He has $50,000 to invest. Because of the risk, he will limit his investment in oil leases and bonds to 30% and his investment in oil leases and stock to 50%. How much should he invest in each to maximize his return, assuming investment returns are as expected? a. Define the variables. Be specific with descriptive words. b. Clearly state the constraints (all inequalities) related to the feasible region. c. State the objective function. d. Set up the initial simplex matrix needed to solve the linear programming problem using the Simplex Method. e. Perform all pivots necessary using row operations to transform the matrix until the solution is feasible.

Using the surface area formula - find the surface area of a sphere.

Let A be a m x n matrix with m<n. Show that (a) If A has a right inverse, then A is full-rank. (b) A cannot have a left inverse.