Matrices

Question 1

How many rows and columns does the matrix A_n×m have?

In other words, which of n and m define the row and column

Answer 1

n rows
m columns

Question 2

What does a_ij represent?

Answer 2

An element at the row i and column j

Question 3

Answer 3

Question 4

What is another word for a matrix with one row?

Answer 4

A row vector

often denoted by lower-case letters in bold type

Question 5

What is another word for a matrix with one column

Answer 5

A column vector

often denoted by lower-case letters in bold type

Question 6

What is another word for a matrix with one row and one column?

Answer 6

A number/scalar

Question 7

What is a zero matrix?

Answer 7

A matrix in which all the entries/elements are 0

Sometimes written as 0_m×n

Question 8

What is a square matrix?

Answer 8

A matrix with equal number of columns and rows

Question 9

What two conditions make the two matrices A=(a_ij) and B=(b_ij) equal?

Answer 9

Question 10

True or false?

Matrices must have the same size to execute addition and subtraction

Answer 10

True.

Two matrices cannot be added nor subtracted if they are of different sizes

Question 11

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Question 12

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Question 13

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Question 14

True or false?

Matrix addition/subtraction is both commutative and associative

Answer 14

True.

Question 15

True or false? GIven a matrix A

A + 0_ij = A

Answer 15

True.

Any matrix added to a zero matrix is still the original matrix

Question 16

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Question 17

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Question 18

GIven matrices A and B

When can you multiply these matrices?

Answer 18

The product of AB exists if and only if the number of columns in A equals the number of rows in B

Note that the product will have the same number of rows as A and the same number of columns as B

Question 19

Answer 19

• AA is possible
• CA is not possible
• AC is possible
• BB is not possible

Question 20

Answer 20

Question 21

Answer 21

Question 22

True or false?

Matrix multiplication is both commutative and associative

Answer 22

False. Matrix multiplication is only associative.

Meaning it does not matter how you bracket it

Question 23

True or false? Matrix multiplication is distributive

Answer 23

True.

Question 24

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Question 25

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Question 26

How would you transpose a matrix?

Answer 26

Swapping the row with the column and the column with the row | Usually denoted by A^T

Question 27

Answer 27

Question 28

# True or false? (A+B)^T=A^T+B^T

Answer 28

True

Question 29

# True or false? (AB)^T=A^TB^T

Answer 29

False. But the following is true

Question 30

What is an identity matrix

Answer 30

A **sqaure** matrix that when multiplied **on either side** of a matrix, A will give A

Question 31

What does an identity matrix look like?

Answer 31

A diagonal line drawn down from top left to bottom right containing only 1's. Everywhere else is 0

Question 32

# Given A is an m×m matrix What is A⁰ equal to?

Answer 32

It is equal to the identity matrix with m rows and m columns

Question 33

Answer 33

Question 34

What is an diagonal matrix?

Answer 34

A square matrix where everything but the diagonal is non-zero ## Footnote The identity matrix is an example of a diagonal matrix

Question 35

How would you calculate the power of diagonal matrices?

Answer 35

Just power the components

Question 36

What does it mean for a matrix to be non-singular?

Answer 36

A matrix, A is non-singular if it is square and there exists a matrix B such that both AB and BA equal the identity matrix I. | Non-singular is a synonym for 'invertable'. B is the inverse of A

Question 37

# Given a matrix A How would you represent the inverse matrix of A?

Answer 37

A^-1

Question 38

Answer 38

Question 39

# True or false? Not every square matrix is invertible

Answer 39

True

Question 41

Answer 41

Question 42

# True or false? (A^-1)^-1=A

Answer 42

True

Question 43

# True or false? (AB)^-1=B^-1A^-1

Answer 43

True

Question 44

# What is another way to write the following? (A^T)^-1

Answer 44

(A^-1)^T | Switching the T and -1 positions

Question 45

# Considering a 2×2 matrix How would you know if it is invertible?

Answer 45

ad-bc≠0

Question 46

# Considering a 2×2 matrix, A and you know A is invertible How would you find the inverse of A?

Answer 46

Question 47

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Question 48

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Question 49

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Question 50

What is the determinant?

Answer 50

A number that determines if a square matrix is invertible or not. | Usually denoted as det(A) or |A|

Question 51

Calculate the following determinant

Answer 51

Question 52

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Question 53

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Question 54

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Question 55

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