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(Marks)
(4) 1.Solve the following system of equations:
2 x 1 − 4 x 2 − 2 x 3 + 8 x 4 = − 4
− 3 x 1 + 4 x 2 − x 3 − 2 x 4 = 0
− x 1 + 3 x 2 + 3 x 3 − 9 x 4 = 5
(4) 2.Set up but DO NOT SOLVE a system of equations which would allow you to find the equation for the
degree 2 polynomial passing through the points (2, 1) and (− 1 , 11) and which has a slope of 3 when
x = 2.
(6) 3.Given that A =
(a) Find the inverse of A.
(b) Use the adjoint formula and the fact that |A| = −1 to find the adjoint of A.
(5) 4.Given the following matrix: A =
0 k 0 1
0 4 0 k
(a) Find |A| in terms of k.
(b) For what values of k is A non-invertible?
(5) 5.Let A be a 4 × 4 matrix with |A| = −3. Let B be a 4 × 4 non invertible matrix.
For each part, either provide an answer or write “not enough information”.
(a) What the value of | 2 A|?
(b) What is the value of |AB|?
(c) What is the value of |A + B + I|?
(d) What is the value of |(A
T A)
− 1 |?
(e) If M is the reduced row echelon form of A, what is the value of |M |?
(6) 6.Given the following matrix: A =
(a) Write A as the product of a lower triangular matrix L and an upper triangular matrix U.
(b) Find elementary matrices E 1 , E 2 and E 3 such that E 3 E 2 E 1 A = U.
(4) 7.Let A and B be n × n matrices such that AB is its own inverse i.e. (AB)
− 1 = AB.
(a) Which of the following is the inverse of BAB (circle your answer)?
(i) ABA (ii) AB (iii) A (iv) BAB (v) BA (vi) B
(b) Is the matrix B necessarily invertible? Justify your answer.
(c) Evaluate and simplify (AB + I)(AB + I).
(d) What is (AB + I) 28 ?
(8) 8.Let T : R
3 → R
2 be defined by T
x
y
z
3 z
2 x − y
and let L be the line
(^) + t
(a) Find the standard matrix for the transformation T.
(b) Sketch the image of the line L under T.
(c) Is T 1-1? onto? Justify your answer in each case.
(d) Let L 0 be the line defined by
b
(^) + t
a
2
. For what a, b does L 0 define the same line as^ L.
(Marks)
(4) 9.Use Cramer’s Rule to find x 3 only in the following system:
5 x 1 − x 2 − 2 x 3 = 7
− 4 x 1 − 4 x 2 + 7 x 3 = 4
− x 1 − x 2 + 2 x 3 = 1
(4) 10.Fill in each blank with the appropriate word. In each case, the appropriate word is either must, might
or cannot. No justification is required.
(a) If A is an n × n matrix such that det(A) = 0, then the system of equations A~x = ~ 0
have a solution.
(b) If B is a set of three linearly independent vectors in P 2 (the vector space of all polynomials of degree
less than or equal to 2) then B be a basis for P 2.
(c) If ~u and ~v are vectors in a vector space S then 3~u − 5 ~v also be a vector in S.
(d) If a transformation T : R m → R n is onto, then there be a non-zero vector ~x such
that T (~x) = ~0.
(8) 11.Find a specific example of each of the following:
(a) A 3 × 3 matrix with every entry different such that |A| = 0.
(b) Two orthogonal vectors in R
3 that have no zero entries.
(c) Two 2 × 2 matrices A 6 = 0 and B 6 = 0 such that AB = 0.
(d) A 2 dimensional subspace of the vector space P 2.
(7) 12.Given that A =
row reduces to
R =
(a) Find a basis for the column space of A.
(b) Find a basis for the row space of A.
(c) Find a basis for the null space of A.
(d) What is rank(A)?
(e) What is dim(Nul(A))?
(f) What is rank(A T )?
(g) What is dim(Nul(A
T ))?
(6) 13.Let H =
{[
x
y
]
: |x| = |y|
be a subset of R
2 .
(a) Does H contain the zero vector of R
2 ? Justify.
(b) Is H closed under vector addition? Justify.
(c) Is H closed under scalar multiplication? Justify.
(d) Is H a vector subspace of R
2 ? Justify.
(5) 14.Let H be the set of all 2 × 2 matrices such that the sum of the entries in H is 0.
(a) Give an example of an invertible matrix which belongs to H.
(b) Find a basis for this subspace of M 2 × 2.
(c) What is the dimension of H?
(3) 15.Find the point of intersection between the plane 3x − 2 y + 5z = 3 and the line x =
(^) + t
