Linear Algebra Answers

Given that M = 2 −1
−3 4
and that M^2 − 6M + kI = 0, find k.

If A =

(i 0
0 i)
, where i =
√−1, find A^2
and A^4
(Matrices)

If A =

(i 0
0 i)
, where i =
√−1, find A^2
and A^4
(Matrices)

13. A and X are the matrices
a b
c d !
and
x y
u v !
respectively, where b is not equal
to zero. Prove that if AX = XA then u = cy/b and v = x + (d − a)y/b. Hence prove
that if AX = XA then there are numbers p and q such that X = pA + qI, and find
p and q in terms of a, b, x, y.

W={(x1, x2, x3)∈R3: x2+x3=0}. Find two subspace W1,W2 of R such that R3= W⊕W1 and R3=W⊕W2 but W1≠W2

if matrix A =
{5 3 -2}
{4 2 1}
{7 -1 4}
a) Multiply A by the matrix
1
1
1
(i)Explain the effect on the elements of A by this product of matrices?

b) (i) Multiply 1 1 1 by A
(ii) Explain the effect on the elements of A by this product of matrices?

Tin Tan Lollies makes three mixes of popular sweets:
The fruit mix contains 50% fruit jellies, 25% chocolates and 25% chews
The chocolate mix contains 30% fruit jellies, 45% chocolates and 25% chews.
The Lasting mix contains 40% fruit jellies, 25% chocolates and 35% chews.
Tin Tan Lollies buys the fruit jellies for $6 a kilogram, the chocolates for $10 a kilogram and the
chews for $5 a kilogram.

a) Write a 3 × 3 matrix, M, that represents the percentage of the three types of sweets in the three
mixes. Convert the percentage in decimal form in matrix M.

b) Write a 3 × 1 matrix C that represents the cost per kilogram of the three types of sweets.

F, C and L represent the cost price in dollars per kilogram of the Fruit, Chocolate and Lasting mixes
of sweets, respectively.
Use the matrix equation MC = P where P = F
C
L

to calculate the values of F, C, L. Give your answers to the nearest cent. Show entire working?

Solve the following national income models by Cramer’s rule Y=C+I+1000,C=10+0.7(Y-T),I=100+0.2Y,T=0.3Y Where Y , C , I, M and T are national income, consumption, investment imports and taxes.

Western Air sold twenty-seven airline tickets to Cape Town on a certain day. Each Economy class ticket (E) was sold at N$3200 and the First-class tickets (F) were sold at N$4100 each. The total amount received for the twenty-seven tickets was N$92700.
4.5.1 Formulate the above information into two linear equations [2]
4.5.2 Solve these equations simultaneously using the substitution method and calculate the number of first-class tickets sold. [6]

Let A = \(\\mathrm{\\{}\)1,2\(\\mathrm{\\}}\), B = \(\\mathrm{\\{}\)a,b,c\(\\mathrm{\\}}\), C = \(\\mathrm{\\{}\)c,d\(\\mathrm{\\}}\). Find \(A\\ \\times\\ (B\\cap\\ C). \)
a.\\(\\left[\\begin{array}{cc} 1 & 0\\\\ 0 & 1 \\end{array}\\right]\\)
b.\\(\\left[\\begin{array}{cc} 1 & 1\\\\ 1 & 1 \\end{array}\\right]\\)
c.\\(\\left[\\begin{array}{cc} -1 & 1\\\\ 1 & -1 \\end{array}\\right]\\)
d.\\(\\left[\\begin{array}{cc} 0 & 1\\\\ 1 & 0 \\end{array}\\right]\\)