Linear Algebra Answers

3. If A = i 0

0 i


where i =√−1, find A^2

and A^4.

Given that M =

2 −1

−3 4

and that M2 − 6M + kI = 0, find k

(Under matrix )

Consider A\\ =\\left[\\begin{array}{cc} 2 & 3\\\\ 1 & 2\\end{array}\\right],\\ B\\ =\\left[\\begin{array}{cc} 1 & 5\\\\ \\end{array}\\right]; what is the identity element of the two matrices

a.\\(\\left[\\begin{array}{cc} 0 & 1\\\\ 1 & 0 \\end{array}\\right]\\)

b.\\(\\left[\\begin{array}{cc} 1 & 0\\\\ 0 & 1 \\end{array}\\right]\\)

c.\\(\\left[\\begin{array}{cc} 0 & 0\\\\ 1 & 1 \\end{array}\\right]\\)

d.\\(\\left[\\begin{array}{cc} 2 & -2\\\\ -13 \\end{array}\\right]\\)

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?