Lecture 1 - Matrix Algebra

Question 1

What is the layout of the subscript when writing the general format for a matrix?

Answer 1

Row first then column

Question 2

How do you add 2 matrices?

Answer 2

Simply add each value with the corresponding value in the other matrix - in the corresponding same position

Question 3

How do you multiply a scalar with a matrix?

Answer 3

Note scalar is just a number

Multiply the scalar with each quantity in the matrix

Question 4

How do you transpose a matrix?

Answer 4

You write each column in the matrix as a row and each row as a column

Question 5

How can you denote a transposed matrix?

Answer 5

A’ - note that a matrix is normally denoted as a capital and bold letter but with a transposed matrix you simply add an aprostophe

Question 6

How do you denote a normal matrix?

Answer 6

Using a capital and bold letter e.g. A

Question 7

How do you denote a vector?

Answer 7

A lower case bold letter e.g. a

Question 8

How do you denote a scalar?

Answer 8

Not bold and lowercase e.g. a

Question 9

What is a matrix?

Answer 9

A set of number organised in rows and columns e.g.:

2 4
6 3

Question 10

Transpose the matrix:

2 4
3 6

Answer 10

2 3
4 6

Question 11

Transpose the vector:

3
4
1

Answer 11

3 4 1

Note that the initial vector is a column vector and transposed it becomes a row vector
It initially can be denoted as a and as a’ after it has been transposed

Question 12

How do you describe/define a matrix?

Answer 12

m x n - where m is the number of rows and n is the number of columns

Question 13

How would you describe/define a transposed matrix?

Answer 13

n x m - interchanging the does and columns

So for example if matrix had 3 rows and 4 columns it could be defined at as a 3 x 4 matrix but once it has been transposed it can be described as 4 x 3 matrix with 4 rows and 3 columns

Question 14

How would you multiply two matrices?

Answer 14

You would multiply the first row of the first matrix by the first column of the second matrix and get the total (which would give you the top left figure)

You would then multiply the first row of the first matrix by the second column of the second matrix and get the total (which would give you the top right figure)

You would then multiply the second row of the first matrix by the first column of the second matrix and get the total (which would give you the bottom left figure)

You would then multiply the second row of the first matrix by the second column of the second matrix and get the total (which would give you the bottom right figure)

Question 15

Multiply the following two matrices together:

1 2 5 0
3 4 1 2

Answer 15

5 + 2 = 7
0 + 4 = 4
15 + 4 = 19
0 + 8 = 8

7 4
19 8

Remember to multiply each figure with its corresponding figure only

Question 16

What is the rule when it comes to adding matrices?

Answer 16

You can only add matrices with the same number of rows and columns therefore and m and n need to be the same respectively

Question 17

What is the rule when it comes to multiplying matrices?

Answer 17

You can only multiply matrices when the number of columns of the first matrix is equal to the number of rows the second matrix
e.g. you can multiply the following two matrices: 2 x 4 and 4 x 1 but you cannot multiply 2 x 2 and 3 x 2 together
I.e the inner two numbers of matrices must be the same for you to multiply them

This is because you a multiplying the rows of the first matrix with columns of the second as therefore need the same number of rows and columns between matrices to allow for all digits to be used in multiplication

Question 18

How can you define/describe the layout of 2 matrices you’ve multiplied together?

Answer 18

Take the two outer numbers of the matrices you’re multiplying to get the layout/definition/description of the new product matrix

E.g. if matrix 2 x 4 was multiplied by matrix 4 x 2 you would get a matrix which can be defined as 2 x 2 I.e a matrix with 2 rows and 2 columns respectively

Question 19

State the layout the product matrix when a 3 x 5 matrix is multiplied by a 5 x 2 matrix

Answer 19

You would get a matrix which can be defined as 3 x 2 i.e a matrix with 3 rows and 2 columns respectively

Question 20

How would you multiply 2 vectors together?

Answer 20

Same way you would multiply a traditional matrix but here you end up with a single scalar quantity as your product

You are multiplying a (1 x n) matrix with a (n x 1) matrix - so your first vector has one row and your second vector has one column - therefore your first vector is a row vector and your second vector is a column vector

But remember n is the same here so the number columns or rows of the first vector will be equal to the opposite of the second vector

Question 21

Multiply the following 2 vectors

4
2 2 4 6
1

Answer 21

8 + 8 + 6 = 22

Note 22 is a scalar

Question 22

How would you calculate (A + B)’?

Answer 22

You can either add matrices A and B as they are first and then transpose the added matrix
Or you can transpose both A and B first so that they become A’ and B’, respectively, and then add them to get your final matrix

Question 23

How would you calculate (A’)’?

Answer 23

A matrix transposed and then re transposed is just it’s the same matrix in its original format so in this case it is just A

Question 24

How would you calculate (lambdaA)’?

Answer 24

Lambda stands for a scalar which cannot be transposed so you simply multiply your lambda (scalar) by the transposed A

Question 25

How would you calculate (AB)’?

Answer 25

Here you would multiply B transposed with A transposed which is: (B’A’)

Note order here matters as you need to multiply the B’s rows by A’s columns

Question 26

What is an identity matrix and what are its features and qualities?

Answer 26

A matrix, denoted as In (n in subscript) where the matrix’s diagonal terms are 1 and the rest are 0 (therefore creating a triangle of 0’s in the top right and bottom left of the matrix)

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

Above is an example of what the identify matrix could look like but it can have n number of columns and rows (therefore n could be any number as long as the number of column and rows are the same)

Note this matrix is both square and symmetric

Question 27

How do treat an identity matrix, denoted at In (n in the subscript), when it is multiplied by matrices A and B of order (m x n) and (n x p) respectively?

Answer 27

Treat it as the number 1 so that when it is multiplied by either A or B it simply cancels away and you are left with either A or B respectively

I.e AIn (n in subscript) = A
And lnB = B

Question 28

What is a square matrix?

Answer 28

A square matrix is when a matrix has the equal number of rows and columns - where m = n

E.g.:
1 3
4 2

Question 29

What is a symmetric matrix?

Answer 29

When a transposed matrix is the same as the ordinary matrix

E.g.:

1 2
2 1

Question 30

What must you remember when multiplying matrices?

Answer 30

Remember that the order in which you multiply 2 matrices matters - as you are multiplying the first matrix’s rows with the second matrix’s columns therefore AB doesn’t equal BA

Question 31

How does the inverse work for matrices?

Answer 31

You can only find the inverse for square matrices and will at most have only one inverse

Matrix B is the inverse of A if AB = BA = ln (identity matrix - n in subscript)

Question 32

State an example for an inverse of a matrix

Answer 32

A:
a11 0
0 a22

A^-1:
1/a11 0
0 1/a22

If you multiplied the 2 matrices above you would get:
1 0
0 1
Which is a identify matrix, In - note you get the same answer even if you were to multiply the 2 matrices the other way round which proves that it is the true inverse

Note the a11 stands for vector in row 1 and column 1 and a22 stands for vector in row 2 and column 2

Question 33

Why is the identify matrix denoted as In?

Answer 33

I stands for identity matrix and n in the subscript stands for the number of rows and columns of the matrix which can vary - there is only one letter needed in the subscript and not two like when describing other matrices because the number of rows is equal to the number of columns

Question 34

When can we use knowledge about inverses?

Answer 34

When solving a system of equations

Question 35

How can we use inverses to solve a system of equations?

Answer 35

Say you have equation Ax = b

And you are told that A is a square matrix and b is a vector

You would first write the equation in its full form which may look like:

1 2 x1 b1
3 4 x x2 = b2

If you multiply the square matrix with x1 x2 you get: x1 + 2x2
3x1 + 4x2

Note that 2 x 2 matrix is being multiplied by a 2 x 1 vector and therefore the product will be 2 x 1 which it is

By multiplying the original equation (Ax = b) by the inverse of A on both sides you get: In x = A^-1b

Question 36

How would you calculate/simplify/rewrite (A^-1)^-1?

Answer 36

The equation is basically saying what is the inverse of the inverse of A and because they two inverses cancel out the answer would just be A

Question 37

How would you calculate/simplify/rewrite (A’)^-1?

Answer 37

The equation is the inverse of A transposed and this is the same and can also be written as (A^-1)’ - therefore the transpose aprostophe and inverse -1 are interchangeable

Question 38

How would you calculate/simplify/rewrite (AB)^-1?

Answer 38

This is the inverse of matrix A multiplied by matrix B - when you have the inverse of two matrices being multiplied together you can rewrite the equation but need to remember to switch the order of multiplication of the matrices therefore the answer would be B^-1 x A^-1

Question 39

How would you calculate/simplify/rewrite (BA)^-1?

Answer 39

Again same rule applies that you need switch the order of multiplication when finding the inverse of two matrices multiplied together therefore your answer would be A^-1 x B^-1

Question 40

How could you calculate/simplify/rewrite (AB)^-1)’?

Answer 40

You can switch the transpose aprostophe and the inverse -1 around and the equation will still be the same ( AB)’)^-1 (see earlier cards for rule)

You can make it (B’A’)^-1 using a rule in earlier cards

Then you need the order of multiplication due to the inverse operation which will make the equation (A’)^-1(B)^-1

Question 41

What is the determinant, why is it used and how do you calculate it?

Answer 41

For a matrix:
a b
c d

The determinant is IAI = ad - bc

Multiply the 2 diagonals and take them away
Determinant written with two vertical lines on either side of a capital A
It shows the variation in the matrix

Question 42

What are some useful rules to remember about determinants for square matrices with order n (therefore n is equal to the number of rows which is the same as the number of columns)?

Answer 42

1) The determinant of a matrix and it’s transposed determinant are the same - try it (when you switch around rows and columns): IA’I = IAI
2) The determinant of a matrix after it has been multiplied is the same as multiplying the determinants of each of matrices being multiplied - again try it: IABI = IAIIBI
3) When a scalar is being mulpltied by a matrix and you want to find the determinant, put the scalar to the power of the number of rows and columns (which should be the same as this is for square matrices) and multiply that by the determinant of the matrix: IyAI = y^nIAI (y here supposed to be lambda which is just a scalar/number)
4) Under collinearity, the determinant is 0 - essentially means that when a row or column can be expressed as the other (collinearity) the determinant will be 0
e.g. 1 1
2 2
Here the determinant is 2-2=0

Question 43

What does it mean for a matrix to be invertible?

Answer 43

A matrix, A, is invertible if and only if the determinant, IAI, is not equal to 0 - I.e there is no collinearity at play

Question 44

What is linear independence?

Answer 44

I think same as collinearity but for vectors instead of matrices

Essentially means that a set of vectors are linearly independent if none of them can be expressed as a linear combination of the other
E.g. 2 and 444 are not linearly independent
2
2

Question 45

What is orthogonality?

Answer 45

When you multiply two vectors together and the product is equal to 0

Remember when you write two vectors being multiplied the first will have an ‘ to indicate it has been transposed because remember that 1 x n vector can only be multiplied by a n x 1 vector - therefore a row vector can only be multiplied by a column vector not another row vector

Therefore consider two vectors x1 and x2
Orthogonality occurs if:
x1’ * x2 = 0

Question 46

What is rank in matrices?

Answer 46

Given a matrix A, the row rank of A is the number of linearly independent row vectors within the matrix and the column rank of A is the number of linearly independent column vectors again within the matrix

Question 47

What is the column and row rank of the identity matrix of order 3?

Answer 47

Identify matrix of order 3 means 3 rows and columns since the matrix is a square matrix therefore:
1 0 0
0 1 0
0 0 1

Here the rows and columns are linearly independent and there is no collinearity therefore the column rank and the row rank are both equal to 3 I.e no row or column can be used to get another row or column respectively

Question 48

What does the symbol with a two sided horizontal arrow (double sided arrow) mean?

Answer 48

If and only if

Question 49

What are the rank properties?

Answer 49

1) row rank = column rank = rank (A)
Row rank is equal to the column rank which is equal to rank of matrix A

2) rank (AB) <= min(rank (A), rank (B))

3) Given that A is a square matrix of order n, A is invertible (which means that the determinant doesn’t equal 0 which means no collinearity is at play and the vectors within the matrix are linearly independent) if only and if rank (A) = n if and only if IAI =/ 0
Rank (A) = n means that all the rows and columns are linearly independent (no collinearity)
Basically same as what we knew before but now an additional condition has been added as we have been introduced to ranks - I.e these 3 statements are the same

Question 50

Stopped making flashcards from slide 21 (inclusive) so need to make flashcards for remainder of L1

Answer 50

A