Matrices & Determinants Questions and Answers

We have det (A) = det(A^T) for any square matrix A. Prove this property for 2 x 2 and 3 × 3 matrices.

Let A and B be 3x3 matrices, with det A=6 and det B=-9. Use properties of determinants to complete parts (a) through (e) below. a) Compute det AB. det AB = __________ (Type an integer or a fraction.) b) Compute det 5A det 5A= ____________ (Type an integer or a fraction.) c) Compute det B det B= ________________ (Type an integer or a fraction.) d) Compute det A^(-1) det A^(-1) = ___________ (Type an integer or a simplified fraction.) e) Compute det A³ det A³ = ___________ (Type an integer or a fraction.)

The simple row-echelon form matrix: 1 2 0 0 0 0 1 0 0 0 0 4, shows that the system is.... of linear equations (1) an inconsistent system (2) a consistent system (3) non linear system (4) non of the above

Pick the correct statements for the given set. -1 -7 7 0 -6 2 Not a basis of R³. Does not span R³. Linearly independent. All of the above.

Consider the following system of equations. 2x + y=2 y-z= 4 -2x + y + 2 = −4 Find the LU-factorization of the coefficient matrix.

Solve the system of equations graphically. Then classify the system as consistent or inconsistent and the equations as dependent or independent. 4u+v=1 4u=v+7

Solve the following system of equations by elimination x + y = 8 4x + y = 23

Find the (3, 1)-cofactor of A =1 9 -2 10 -8 -8 -6 -5 2 C31=

Find the standard matrix for the linear transformation T on R2 that first projects the resulting vector on y-axis then rotates the resulting vector with an angle, and then finally reflects a vector about the y-axis.

Let B=(b₁ b₂} and C= (C₁.C₂} be bases for a vector space V, and suppose b₁ = -2c₁ +6c₂ and b₂ = 8c₁ - 3c₂ a. Find the change-of-coordinates matrix from B to C. b. Find [x]c for x = 6b₁-5b₂. Use part (a). a. P = C←B

1. A matrix model is given by the following system of linear equations. A(t+1)=2At + Ct B(t+1)=3At + 5Bt C(t+1)=4A +6Bt (a) Write the corresponding matrix equation. (b) Write the corresponding matrix diagram.

Find the product of the two matrices below. Is each the inverse of the other? 7 4 3 4 -5 3 -5 -7 Find the inverse of matrix A by using row operations to get [1] A-¹] A = 4 3 3 2

Let V =x x-y+z= 0 and 2x + z = 0 y ER³ z (a) Find a matrix A such that V = Null(A). (b) Find a matrix B such that V = Col(B). (c) Find a matrix C such that V = Row (C). (d) Determine the geometric nature of V. In other words, determine if V is a line, a plane or all of R³.

Solve the equation 8x-7y-5z = -11. x = +s +t y z

Use the Gram-Schmidt Orthonormalization Process to transform the basis B = {(1, 0, 1), (1, 2, 0), (2, 1, 1)} into an orthonormal basis for R³.

A 3x3 matrix has eigenvalues (3, 2, 1) and (1, 1, 1). Is the matrix diagonalizable? Explain.

Find a basis of the subspace of R³ defined by the equation - 8x1 - 3x2 + 6x3 = 0.

Use matrices to solve the system. 5x-9y-5z = 4 6x-5y + 7z = 77 4x + 7y-8z = -35 Write the solution as an ordered triple in the form (x,y,z). Do not enter any spaces in the answer. The solution is__

Find the vertex and the intercepts of the graph of the function. f(x) = (x + 2)² - 9 Do not enter any spaces in the answers. The vertex is Enter the x-intercepts as two points separated by a comma. Do not enter any spaces. The y-intercept is the point

Let A and B be invertible nxn matrices. Find a formula A-1 in terms of B if: (3ATB − I)-¹ = (A-¹B)T Your formula for A-1 should be simplified and should not depend on A.

Interpret the slope in terms of this application. The annual cost at the community college increased by 366 dollars per year since 2005. The annual cost at the community college increased by 3 dollars per year since 2005 The annual cost at the community college increased by 1098 dollars per year since 2005. The annual cost at the community college decreased by 366 dollars per year since 2005. The annual cost at the community college increased by 1 dollar per 366 years since 2005.

Find the percent change to the nearest percent : 25 feet to 105 feet.

Bill is playing a board game that has a spinner divided into equal sections numbered 1 to 20. The probability of the spinner landing on an even number or a multiple of 5 is? Select one: a. 1/5 b. 7/15 c. 12/20 d. 3/5

Find a 4 x 4 matrix that represents in homogeneous coordinates the rotation by an angle θ about the x = y = 1, z = 0 line of R^3.

Solve the equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 9x+6=7x+6 What is the solution? A. The equation has a single solution. The solution set is . B. The solution set is {x|x is a real number}. C. The solution set is Ø.

Find the exact value of the expression. sin (2 cos -¹ (-12/13))

Let S be the the ellipsoid given by the equation x² + y² + 6z² = 32. Find the biggest and smallest values that the function f(x, y, z) = x+y+6z achieves on the part of S that lies on or above the plane x+7y+6z =

Determine if the following set of polynomials is an independent subset of P₃. S = {1+x+x³,x+x²+2x³,1-x³}

Find both the standard and parametric equations of the plane containing the points (-1,2,3), (0, 1,-1), and (1,0,2).

A quantity with an initial value of 300 decays exponentially at a rate such that the quantity cuts in half every 10 years. What is the value of the quantity after 57 months, to the nearest hundredth?

Find the coordinate matrix of x = (1, 2, 3) relative to the basis a1= (1,1,0), a2 = (0,1,1), and a3 = (1,1,1).

Transition matrix from the basis X = {(1, 2, 3), (1, 0, 1), (1, 2, 1)} to X' = {(1,1,0), (0,1,1), (1, 1,1)}

Define the linear transformation T: Rn⇒Rm by T(v) = Av. Find the dimensions of Rn and Rm. A [-1 0 -1 0] dimension of Rn dimension of Rm

Determine the solution of the simultaneous nonlinear equations y = -x² + x +0.5 y + 5xy = x² Use the Newton-Raphson method and employ initial guesses of x = y = 1.2.

Consider the linear subspace U of R4 generated by {(2,-1,3,-2), (-4,2, -6,4)}. The dimension of U is a) 1 b) 2 c) 3. d) 4

Answer each of the following as True or False justifying your answers: c) The inverse of a non-singular matrix A is unique

Let k ε R and consider the linear system in the unknowns x, y, z: kx + 2y+z=0 y+z=0 3z = 0 Find all values of k for which this system has (a) a unique solution, (b) infinitely many solutions, (c) no solutions.

Find the basis and rank of the vector system. Express non basis vectors with basis vectors: a1=(-2;3;4;11), a2=(7;-2;3;4), a3=(1;-3;-5;-13), a4=(3;2;7;16)

Use matrices to solve the system. 3x +2y-z = 11.6 5x -3y + 2z = -7.7 -2x + y - 2z = 6.7

Use matrices to solve the system. 2w + 2x +2y+z=-2 3w+ 4x + y2z = -10 4w + 3x - 2y + 2z = 22 w - 2x+3x = 7

Find a unit vector that has the same direction as the given vector. v=-5,6

Suppose A € R5x5 has det(A) = 10. Determine det (2A³) and explain your answer.

Given the matrix A 1357. What is the inverse of A? A. -187-3-51 B. 187-3-51 C. -167-3-51 O. 167-3-51

Suppose Amanda, Claudia and Delaney go to a donut shop. Amanda buys 6 apple crumb, 6 brown butter pecan and 2 chocolate frosted donuts and ends up spending $19.56. Claudia gets 5 of the apple crumb, 4 brown butter pecan and 2 chocolate frosted donuts for a total cost of $15.94. Lastly, Delaney spends $19.69 on 5 apple crumb, 8 brown butter pecan, and 3 chocolate frosted donuts. Write this scenario as a system of equations, then solve that system using an augmented matrix and RREF on your calculator before answering the question below. How much does a single chocolate frosted donut cost? Round your answer to 2 decimal places.

Find the linear function, f(x), passing through the points: (-10, 10) and (8, 10) Enter your answer using function notation.

ABC is a triangle. Pis a point on BC such that BP: PC=3:2 and Qis a point on AB such that AQ: QB=1:2. Line AP and CQ intercept at R. Find AR: RP and CR: RQ.

Let A, B be 4 x 4 matrices with |A| = -2 and |B| = 5. Calculate |- 2A¯¹ B² AT |.

Let A = [110] b = [3] Find A^-1 and use it to solve the system Ax = b.

Let V = R⁵ and let W be a the subspace of all (a, 0, b, 2a +b, -3b). Find a basis for W. Select the correct answer form the options below. O S= {(0, 0, 1, 1, -3), (1, 0, 2, 0, -3)} O W does not have a basis. O S = {(1,0,0,2, 0), (0, 0, 1, 1, -3)} O S = {(1,0,0,0,0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 1)}

Consider the following 3 x 3 matrix: [ζ 1 7 ζ 3 2 4 -1 1] Find the values of for which the matrix is: 1. positive definite 2. positive semidefinite 3. negative definite 4. negative semidefinite 5. indefinite