Linear Algebra Answers

check that p^(e)andp^(0) are subspace of R[x] if p^(e)={p(x)€R[x]|p(x)=p(-x) and p^(0)={p(x)€R[x]p(x)=-p(-x)

the height of an above ground pool is 3 feet. the pool needs to be drained. as the water drains, the height of the water changes at a rate of -(1)/(2) inch per minute. write and sold and equation to find how many minutes it will take to drain the pool

The profit made by a company when 60 units of its product is sold is R 1 600.00. When 150 units of its products are sold, the profit increases to R 5 200.00. Assuming that the profit function is linear and of the form

⎛ 2 0 1 ⎞
9.Given that A = , ⎜ k 2 3 ⎟ what is the value of k, if A is said to be a singular matrix?
⎝ 2 1 4 ⎠

a.8
b.10
c.7
d.-6





⎛1 2 3⎞
10.let A= ⎜4 5 0⎟ then the cofactor of matrix A is the matrix
⎝2 1 4⎠,

a.

⎛ 20 −16 −6 ⎞
A = ⎜−5 −2 3 ⎟
⎝−13 12 −41 ⎠

b.
⎛ 20 −16 −6 ⎞
A = ⎜−5 −2 3 ⎟
⎝−13 12 −3 ⎠

c.
⎛ 20 −16 −6 ⎞
A = ⎜−5 −2 3 ⎟
⎝−15 12 −3 ⎠

d.
⎛ 20 −12 −6 ⎞
A = ⎜−5 −2 3 ⎟
⎝−13 12 −41 ⎠

⎛1 2 3⎞
5. Given that A= ⎜3 2 1⎟ . Find the determinant of A
⎝1 3 2⎠


a.2
b.3
c.1
d.zero

6. A matrix is said to be singular if the determinant is equal to
a.3
b.1
c.zero
d.2

3. A necessary and sufficient condition for a matrix (square) A to be invertible is that
a.A is not equal zero
b.|A|≠0
c.|A|>0
d.A<0

4. Given that x + 2y + 3z =1, 3x + 2y + z = 4, x + 3y + 2z = 0. What is x,y and z?
a.(7/4,-3/4,1/4)
b.(5/4,-2/4,1/4)
c.(1,-3/4,1/4)
d.(1,-2,3)

1.Given that x + 2y = 3, 3x + 4y =1. What is x and y?
a.(1,-1)
b.(2,-2)
c.(-5,4)
d.(4,-5)

2.Given that
(1 3)
(k 4)
is singular matrix. Find the value of k
a.(4/3)
b.(3/5)
c.(3/4)
d.(5/3)

Define T : R
3 → R
3 by
T(x, y, x) = (−x, x−y,3x+2y+z).
Check whether T satisfies the polynomial (x−1)(x+1)
2
. Find the minimal
polynomial of T.

Let P superscript (e) ={p(x)∈R[x]|p(x) = p(−x)} P superscript(o) ={p(x)∈R[x]|p(x) =−p(−x)} a) Check that P superscript (e) and P superscript (o) are subspace of R[x]. b) Show that P superscript (e) =(∑ i a subscript i x superscript i ∈R[x]



 a subscript = 0 if i is odd.) P superscript(o) =(∑ i a subscript i x superscript i ∈R[x]



 a subscript = 0 if i is even.) Deduce that P superscript (o)∩P superscript (e) ={0}. ( c) Check p(x)+p(−x)∈P superscript(e) for every p(x)∈R(x). Check that the map ψ: R[x]→P superscript(e) given byψ(p(x)) = p(x)+p(−x)/ 2 is a linear map. Further, check that ψ superscript 2 =ψ. Determine the kernel of ψ

Check whether the matrices A and B are diagonalisable. Diagonalise those matrices which are diagonalisable. i) A =  1 0 0
1 2 −3
1 1 −2  
ii) B =  −2 −4 −1
3 5 1
1 1 2 .b) Find inverse of the matrix B in part a) of the question by using Cayley-Hamiltion theorem. c) Find the inverse of the matrix A in part a) of the question by finding its adjoint.