Matrices & Determinants Questions and Answers
Algebra
Matrices & DeterminantsWhich of the following is the trace below 00 01 10 trace AB7 where A CA 0 0 and 8 2 1 2 B 0 10
Algebra
Matrices & DeterminantsLet C 1 a b b Compute QQ and so find Q if Q 0 Q a b c ba d C d d C c d a
Algebra
Matrices & DeterminantsLet A 123 1 28 and v G Also let H be the two column vectors of A Compute A and A a Av b 44 5A What do you notice about these answers 28
Algebra
Matrices & DeterminantsConsider the linear system 22 12 10 2 a 142 a For which value of a will the system have no solutions b For which value of a will the system have infinitely many solutions case Use t as a parameter for your solution z t y t Write down the set of solutions in this D c In the other cases the system has a unique solution This solution depends on the value of the variable a and is equal to B
Algebra
Matrices & DeterminantsFor k 0 use the row reduction method to find A 1 A if given 12 0 13 1 13 k 1
Algebra
Matrices & Determinants1 point Let T R R be a linear transformation such that 2 0 9 0 4 0 0 T Find the following 7 0 8 3 7 Hint First write 5 3 as a linear combination of 9 400 and 1
Algebra
Matrices & Determinants1 For k 0 use the row reduction method to find A if given 1 2 0 3 1 3 k 1 A 1 1
Algebra
Matrices & DeterminantsLet 1 3 0 1 k k 0 31 Find the values of k for which A is not invertible NOTE Enter all values exactly Number of answers 0 A
Algebra
Matrices & DeterminantsConsider B ir Ma 3 what is Me B 1 and 1 A ve vc B M 2 ve 11 22 0 v c 3 C are bases for R2
Algebra
Matrices & DeterminantsFind a basis for the nullspace of the matrix If there is no basis enter NONE in any single cell 13 6 4 8 6 3 2 4 2 1 1 2 4 2 2 3 A
Algebra
Matrices & DeterminantsLet 4 1 2 B 2 C 113 1 D 3 Evaluate each of the following if possible If it is not possible justify your answer Show all your work CET 3E 13 tr D B Find the first column of BA a 3 marks b 2 marks c 4 marks d 5 marks 1
Algebra
Matrices & DeterminantsConsider A What is the maximal order of a minor A 1 B 3 C 4 5 0 1 5 2 3 4 1 2 1 3 3 1 D 2
Algebra
Matrices & DeterminantsLet A 272 1456 Decide whether the matrix A is invertibile and if so use the adjoint method to find its inverse Enter as a matrix exactly in fractional from if required if not invertible enter NA
Algebra
Matrices & DeterminantsThese points are linear Find the slope x 3 2 1 012 y 3 2 10 12 slope Enter
Algebra
Matrices & DeterminantsLet A A 11 If the matrix is invertible use the adjoint method to find its inverse If the matrix is not invertible indicate that using the checkbox NOTE Write the elements of the matriz exactly 2 2 0 0 0 0 2 1 2 0 10 14 T 121120 3 20 10 0 X
Algebra
Matrices & Determinants1 point Let Define the linear transformation T R R by T A Find the images of u H 4 T T u 0 7 0 BH A 0 02 800 T v 4 and 7 a 1 C b under
Algebra
Matrices & Determinants5 In the diagram the inscribed angle LABC has a measure of 50 AB BC and the radius of the circle shown is 10 units What is the length of AB A 357 16 B 657 6 C 657 9 D 42n 5 B A C
Algebra
Matrices & DeterminantsIf V V2 V3 is a basis of R3 show that U 42 433 is also a basis of R3 U V U V V U3 V V3 2 The column space and null space of a of 7x10 matrix A are subspace the same dimension Find nullity AT
Algebra
Matrices & DeterminantsUse Cramer s rule to solve the system y z 4 X y 2z 14 x 4y 2z 17 X Find the determinants D x 0 0 0 0 Select the correct choice below and fill in any answer boxes within your choice OA The solution set is Simplify your answers OB The system contains dependent equations OC The system is inconsistent www
Algebra
Matrices & DeterminantsProblem 13 7 points Find the eigenvectors for the matrix A B 1 1 9 6 D C C E 4 0 1 03 0 102
Algebra
Matrices & Determinantsc Find a basis for the solution space for AX 0 where A What is the dimension for this solution space 1 2 2 2 1 1 1 1 2 0 1 3 1 3
Algebra
Matrices & DeterminantsA p 2 B F 3 Given A use the Frobenius inner product and the corresponding induced norm to determine the value of each of the following A B and B 5 2 4
Algebra
Matrices & Determinants1 point For which value of k does the matrix A 58 5 6 have one real eigenvalue of multiplicity 2 k
Algebra
Matrices & Determinantsbelow Thanks 1 Confirm by multiplication that is an eigenvector of A and find the corresponding eigenvalue 1 2 1 X 32 a A 4 0 1 b A 2 32 104 X 3 2 1
Algebra
Matrices & Determinants2 Solve the system of algebraic equations x 2x 0 X 2x x 1 3x x x 6 by Cramer s method Check the answer
Algebra
Matrices & Determinantsa Solve the following equation for matrix X 0 0 0 P 000 1 1 0 23 RX B 3 x 2 12 0 1 1 23 12 1 1 1
Algebra
Matrices & Determinants3 Construct an orthonormal basis for the subspace of R4 W span x1 x2 x3 15 x 0 3 4 0 C x2 70 2 1 2 C X3 1 0 0 0
Algebra
Matrices & Determinants1 point Which of the following sets of vectors are linearly independent Check the boxes for linearly independent sets 8 0 A B 0 7 5 1 2 4 2 D 5 3 1 8 4 C 6 5 0 0 0 4 5 1 2 7 4 G E 2 7 8
Algebra
Matrices & DeterminantsWhich of the following is NOT in the span of x and y 2 1 1 1 3 8 2 3 H 3 5 Y
Algebra
Matrices & DeterminantsFind the rank and nullity of the matrix then verify that the values obtained satisfy Formula 4 in the Dimension Theorem 4 69 24 10 11 0 2 4 2 4 5 13 21 30 A 2015 rank A nullity A rank A nullity 4
Algebra
Matrices & DeterminantsFor the encoding matrix 13 2 5 3 10 A 29 54 6 19 decode the message GE 42 109 433 198 181 which gives a six character word 797
Algebra
Matrices & Determinants2 1 13 52 26 find an orthogonal basis under the Euclidean inner product Let 3 112 Orthogonal basis V x a Ex 1 23 113 3 V 3 b Ex 1 23 33 22 26 V3 c Ex 1 23 be a basis for R Use the Gram Schmidt process t a B b
Algebra
Matrices & DeterminantsLet W Span x1 x2 where 1 and 1 3 1 3 2 3 Compute the Orthonormal Basis Using Gram Schmidt Process
Algebra
Matrices & DeterminantsExpress each column vector of BB as a linear combination of the column vectors C C2 and c3 of B 3 4 2 B 0 1 6 5 Enter first column as a linear combination of columns of B in terms of the vectors C C2 and c3 Enter second column as a linear combination of columns of B in terms of the vectors C C2 and c3 Enter third column as a linear combination of columns of B in terms of the vectors C C2 and c3
Algebra
Matrices & DeterminantsUse Gaussian Elimination to carry the following matrix to a REF 2 31 2 4 5 1 1 4 2 3 7 What are the positions of the pivots 1 2 1 2 1 1 1 2 3 and 3 4 1 1 and 2 3 1 1 2 2 and 3 3 1 1 2 2 3 3 and 4 4 1 1 2 2 and 3 4
Algebra
Matrices & Determinants1 Let A a n be symmetric and positive definite Show that a a 0 i 1 2 n b a aa i j 1 2 1
Algebra
Matrices & Determinants1 A network of six one way streets A B C D E F is shown in the following diagram where 100 cars per hour enter at streets A and B 200 cars per hour enter at streets D and E 50 cars per hour exit at streets A and C and 250 cars per hour exit at streets E and F 100 A 200 B 50 C D E 250 a 4 marks Represent the flow of cars in this network using a system of linear equations and represent the system using an augmented matrix b 5 marks Identify the rank of the system and find a possible solution
Algebra
Matrices & DeterminantsWhat is the image of X 1 A 2 32 B 2 under the linear transformation T where I is the reflection across X2 X1
Algebra
Matrices & DeterminantsThe linear transformations T R R and T R R are defined by T1 X EDH X3 T X2 What is T T 2 W A 2 B 7 X1 X3 X2 X3 X3 X1 3 X1 CH X3 X1 X1 X2 X1 x x3 and
Algebra
Matrices & Determinants8 Find the points on the surface z xy 1 that are closest to the origin using LaGrange multipliers
Algebra
Matrices & Determinants7 Solve the inequality set Graph the solution on a n 14 x 5 O x 9 or x 19 O x 9 or x 19 O x 9 or x 19 O x 9 or x 19 O all real numbers no solution
Algebra
Matrices & DeterminantsUsing the discriminant state the number of soluti For the following equation 3y 18 one real root two real roots three real roots O no real roots Using the discriminant state the number of soluti for the following equation 4x x 5 one real root two real roots three real roots
Algebra
Matrices & DeterminantsSolve the system if possible by using Cramer s rule If Cramer s rule does not apply solve the system by using another method Write all numbers as integers or simplified fractions 2x 6y 3z 2 4x 12y 6z 4 6x18y 9z 6 Part 0 2 Part 1 of 2 Evaluate the determinants D D D and D
Algebra
Matrices & DeterminantsUse the Gram Schmidt process to produce an orthogonal basis for the column space of matrix A An orthogonal basis for the column space of matrix A is Tyno a voctor or list of woctore Ilco o con congrato vectors of nooded A 9 1 7 16 1 9 15 16 1 9 23 3 5 6 21 15 13 6 3 O
Algebra
Matrices & Determinants4 The following cyclic subgroups are all isomorphic the multiplicative group i in C the additive group Z and the subgroup 6 1 of S 23 2 3 4