Matrices & Determinants Questions and Answers

Costumes have been designed for the school musical. Costume A requires 5 yards of fabric, 4 yards of ribbon, and 3 packets of sequins. Costume B requires 7 yards of fabric, 4 yards of ribbon, and 1 packet of sequins. Fabric costs $5 per yard, ribbon costs $3 per yard, and sequins cost $1.50 per packet. Use matrix multiplication to find the total cost of the materials for each costume.

For what value(s) of k will the following system of linear equations have no solution? infinitely many solutions? x-3y = 5 - 6x + 18y = k ... Select the correct choice below and, if necessary, fill in the answer box to complete your choice. OA. The given system has no solution for all real numbers k except for k= (Use a comma to separate answers as needed.) OB. The given system has no solution for all real numbers k. OC. The given system has a solution for all real values k.

Mark each statement True or False. Justify each answer. a. A row replacement operation does not affect the determinant of a matrix. b. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)^r, where r is the number of row interchanges made during row reduction from A to U. c. If the columns of A are linearly dependent, then det A = 0. d. det(A + B) = det A + det B.

Solve each system analytically. If the equations are dependent, write the solutions set in terms of the variable z. 3x+6y-z = 6 2x-3y + 5z = -10 8x+9y-2z= 12

Solve the system analytically. If the equations are dependent, write the solution set in terms of the variable z. x+y+z= -2 2x+y-z = -14 x-y+z= -4 Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice. A. There is one solution. The solution set is . B. The system has infinitely many solutions. The solution set is ..., where z is any real number. C. The solution set is Ø.

If the column space of a 3x3 matrix consists of all vectors b=[b₁ b2 b3] such that b₁+ b2= 3b3 then one of the following set of the vectors forms abasis for the left null space of that matrix: [113] and [11-3] [11-3] [331] and [33-1] [33-1]

(10 points) Write the augmented matrix from the system of equa- tions. Then, solve the system using Gaussian elimination (elementary row operations). x - 3y + 2z = 12 2x5y + 5z = 14 x - 2y + 3z = 20

Write a matrix equation equivalent to the following system. 2x - 3y = -1 4x + 5y = 9 Find the inverse of the coefficient matrix, and use it to solve the system.

In the vector space P2, of polynomials of degree at most two, the third column of the matrix of from the transition basis B={e₁=4-x², e ₂ = −2 + 3x², e3=1+x} to the basis A={a₁=1, a₂=x, a3=x²} is:

Use Gauss-Jordan row reduction to solve the given system of equations. HINT [See Examples 1-6.] (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer in terms of z, where = x(z) and y = y(z).) 8x - y + z = 2 9x - y + z = 5 (x, y, z) =

Write the following system of equations as a matrix equation and solve the system by finding the inverse of the coefficient matrix. |3x-y+2z=-1 |4x-2y+z=-7 |-x+3y-2z=-1

If A is an invertible square matrix of order less than 4 such that |A-AT|!=0 and B = adjA, then which of the following is/are always true - adj(B²A-¹B-¹A) = A adj(B²A-¹B-¹A) = B Order of A is 2 Order of A is 3

Prove that any Möbius transformation can be written in a form with determinant 1, and that this form is unique up to sign. How does the determinant of T(z) = (az+b)/(cz + d) change if we multiply top and bottom of the map by some constant k?