Semester 2

DIFFERENTIATION (CALCULUS) EXTENSIONS

DIFFERENTIATION (CALCULUS) EXTENSIONS

Recap: what are all the interesting features of a function? (5 old, 3 new)

1. Vertical axis intercept (y-intercept)

2. Horizontal axis intercepts/roots - use discriminant (b² - 4ac), if no roots then doesn’t go through/touch horizontal axis

3. Stationary points - f’(x) = 0, f’’(x) < 0 or > 0 OR = 0 for POI, f’’’(x) < 0 or > 0 upward/downward flowing

4. Non-stationary points - only for cubic or higher f’’(x) = 0, f’’’(x) < 0 or > 0 upward/downward flowing

5. Continuous - limits

6. Concave or convex - (2nd derivative)

7. Smooth or kinked

8. Constraints - binding or slack

*if there’s no number at the end of the equation, e.g. y = 3x + 2x

GROWTH, DISCOUNTING AND NATURAL NUMBERS

GROWTH, DISCOUNTING AND NATURAL NUMBERS

MATRIX ALGEBRA

MATRIX ALGEBRA

In a matrix, what order do we have to read it in first to get the position of any element?

d

• Matrix bold (collection of numbers)

• Elements not bold (only one element, not a collection)

• Can write elements with comma or without - e.g. A₅₂ OR A_5,2

What are the 6 different types of matrix’s?

1. Vector matrix - column matrix

2. Vector matrix - row matrix

3. Square matrix

4. Square matrix - identity matrix

5. Null (zero) matrix

6. Scalar

For the 2 types of vector matrix’s:

• What is the special characteristic for both?

• How do you denote a vector matrix?

• One of the dimensions is EQUAL TO 1

• A vector is denoted as a lower case bold

For the 2 types of square matrix:

• Describe both and how they are different

Describe a:

• Null (zero) matrix

• Scalar

*don’t ever really see null matrix*

Scalar = 1×1

What are the 3 different matrix operations?

1. Transpose (not used that often)

2. Addition/subtraction

3. Multiplication

What is transpose?

• “Flip things”

• Denoted by a prime - e.g. A’ or b’

• Exchanges the row and column position of the elements in the matrix - so A (3×2) becomes A (2×3)

• E.g. position A₁₂ = A’₂₁

How is transpose different for:

• a square matrix

• a (square) identity matrix

Matrix’s where the transpose is the same (D=D’) = symmetric matrix

> ALL identity matrix are symmetric

> Has to be square to be a symmetric matrix

What are the conformability conditions for:

• Addition/subtraction

• Multiplication

Addition/subtraction: must be SAME DIMENSIONS

> e.g. A + B = 3×3

Multiplication: number of columns in 1st matrix = number of rows in 2nd matrix

> e.g. A = 3×2, B = 2×3 (2=2)

Are addition/subtraction & multiplication:

• Commutative (swap position of numbers and still get the same answer)

• Associative (grouping of numbers change and still get same answer)

Addition/subtraction:

• Commutative > e.g. A + B = B + A

• Associative > e.g. (A + B) + C = B (A + C)

Multiplication:

• NOT commutative >

How do you add/subtract a matrix?

Elements in the same position are added/subtracted to form the resulting matrix

What happens when you add/subtract a square matrix and its transpose together?

Gives you a symmetric matrix

What happens if you add a matrix to a null matrix?

Leaves initial matrix unchanged (null = 0)

What are the steps to multiplying matrices:

• Vectors/Square matrix

• Scalars

Vectors/square:

1. Check for conformability - e.g. if A = 3×2 and B = 2×3 then this is conformable, if not then can’t multiply

2. Work out dimensions of resulting matrix - by looking at rows in 1st matrix & columns in 2nd matrix, e.g. 3×3 in this case

3. Draw out each position of new matrix - *optional for just makes it easier*, e.g. B11, B12, etc.

4. Multiply + add each position - e.g. B12 = row 1 (matrix 1) column 2 (matrix 2) > for each element multiply its corresponding element then add it up to get total of B12

Scalars:

1. Times everything by scalar - e.g. [ 2 3 8 ] and scalar = 2 then it would be [ 4 6 16 ]

What is a determinant?

• What type of matrix can you work out its determinant for?

• How is this denoted?

SINGLE NUMBER

> only for square matrix - denoted by ‘absolute value’

e.g. V 2×2 = |V|

How do you work out the determinant for a (square) matrix which is a 2×2?

1. Subtracting diagonals - subtract product of the back diagonal elements, from the product of the forward diagonal elements, e.g. (V11xv22) - (V12-V21) = |V| *LEFT TO RIGHT*

2. Classify matrix - determine whether its:

• Non-singular (any positive or negative number)

• Singular (0) - has some linear dependency between rows and columns (collinearity)

3. Rank - it’s the number of linearly independent columns (or rows)

> If rank < columns then determinant = 0

> If rank = columns then determinant = non-singular

How do you work out the determinant for a (square) matrix which is a 3×3 or more?

Matrix multiplication = addition

Matrix determinant = subtraction

Matrix multiplication = addition

Matrix determinant = subtraction