# Matrices & Determinants Questions and Answers

Algebra
Matrices & Determinants
If A is a m x m matrix and rank (A - Im) = m - 1. Show that is an eigenvalue of A with multiplicity of at least one.
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
Let A = 2 9 -2 6 1 6 -1 4 Find an orthonormal basis of the kernel of A.
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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?
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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)
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
x - 2y = 23/2 x + 3y = 33/2 Solve Using a Matrix with Cramer's Rule
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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) =
Algebra
Matrices & Determinants
4. Find the equation of an ellipse passing through (-4,0) with foci at (0,0)and (-1,4).
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
Let R* be equipped with standard inner product. Find all vectors in R* orthogonal to v = (1, -5,8,2)
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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)
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
What is the scale factor if a 8" by 16" photograph is enlarged to a poster that is 2 ft. by 4 ft.?
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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) =
Algebra
Matrices & Determinants
Determine whether S = {1-t,2t+3t2,t²-2t³,2+t³) is a basis for P₁.
Algebra
Matrices & Determinants
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?
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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.)
Algebra
Matrices & Determinants
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?
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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)
Algebra
Matrices & Determinants
Find the following for the function f(x)=2x+3/7x-8
Algebra
Matrices & Determinants
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
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
Determine the integers n for which the pair of congruences 2x-y = 1 and 4x + 3y = 2 modulo n has a solution.
Algebra
Matrices & Determinants
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?
Algebra
Matrices & Determinants
Let A = A is a 4X4 matrix . Show that A is invertible by Gershgorin Circles.
Algebra
Matrices & Determinants
Let A = A is a 4X4 matrix. Show that A is invertible by Gershgorin Circles.
Algebra
Matrices & Determinants
Let A be a symmetric 4 x 4 matrix such that det(A) = 3. Find adj(3A)A^T.
Algebra
Matrices & Determinants
Let B be a symmetric 5 x 5 matrix such that det(B) = 2. Find adj(2B) BT
Algebra
Matrices & Determinants
Let A be a symmetric 4 x 4 matrix such that det (A) = 3. Find adj (3A) AT.
Algebra
Matrices & Determinants
Use Gauss-Jordan elimination to solve the following system, 3x +9y+ 2z + 12w= 1 x + 3y - 2z + 4w= 2. -2x - 6y - 10w= 0,
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
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.)
Algebra
Matrices & Determinants
Show that every matrix "A" with real entries can be same as symmetric and skew symmetric matrix
Algebra
Matrices & Determinants
If A and B are invertible, then AB and BA are similar. (a) True (b) False
Algebra
Matrices & Determinants
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.
Algebra
Matrices & Determinants
Using the surface area formula - find the surface area of a sphere.