Matrices

Question 1

# define:
matrix

Answer 1

A

• *rectangular arrangement** of
• *numbers** into
• *rows and columns**.

Question 2

In plainspeak,
what is a
matrix?

Answer 2

A
compact representation of numbers.

Question 3

What are the
dimensions
of matrix B?

Answer 3

3 x 2

Pronounced “three by two.”

Rows x Columns

Question 4

• *Envision** a
• *2 x 3 matrix**.

Answer 4

Rows x Columns

Question 5

What is

• *another word** for a
• *matrix entry**?

Answer 5

Matrix element

Question 6

How could you

• *identify** the
• *entry −7** in
• *matrix G**?

Answer 6

g_1,3

It’s the entry in the first row and the third column.

Question 7

What is
element g_2,1?

Answer 7

18

It’s in the second row and the first column.

Question 8

# _define_:
augmented matrix

Answer 8

A matrix that

• *represents** a
• *system of equations**.

Question 9

In an
augmented matrix,
what does each
row represent?

Answer 9

One equation
in the system of equations

Question 10

In an
augmented matrix,
what does each
column represent?

Answer 10

A
variable
or the
constant terms
in the system of equations

Question 11

• *Envision** the
• *augmented matrix** that
• *represents** the
• *system below**.

Answer 11

Question 12

• *Envision** the
• *augmented matrix** that
• *represents** the
• *system below**.

Answer 12

Question 13

• *Envision** the
• *augmented matrix** that
• *represents** the
• *system below**.

Answer 13

Question 14

What are the

• *three** elementary
• *matrix row operations**?

Answer 14

Switch any two rows

Add one row to another

Multiply a row by a
nonzero constant

Question 15

Why can you
switch any two rows in an
augmented matrix?

Answer 15

The

• *order** of the equations
• *doesn’t matter**.

Question 16

Why can you
add one row to another in an
augmented matrix?

Answer 16

Because you can

• *add two equal quantities** to
• *both sides** of an equation.

Question 17

Why can you
multiply a row by a nonzero constant in an
augmented matrix?

Answer 17

Because you can

• *multiply both sides** of an equation by the
• *same nonzero constant**.

Question 18

How do you

• *notate**
• *interchanging rows 1 and 2**?

Answer 18

Question 19

How do you

• *notate**
• *multiplying row 2 by three**?

Answer 19

Question 20

How do you

• *notate**
• *replacing row 2** with the
• *sum of rows 1 and 2**?

Answer 20

Question 21

Answer 21

Just add the corresponding entries.

Question 22

Answer 22

Just subtract the corresponding entries.

Question 23

Answer 23

Undefined

Cannot add or subtract matrices with different dimensions.

Question 24

When working with
matrices,
how do you
refer to
real numbers?

Answer 24

Scalars

Any real number that is
not a part of the matrix is a
scalar.

Question 25

# _define_:
matrix equation

Answer 25

An

• *equation** in which the
• *variable** stands for a
• *matrix**.

Question 26

Answer 26

Question 27

# _define_:
zero matrix

Answer 27

A

• *matrix** in which
• *all entries are 0**.

Question 28

When working with
matrices,
what does
O mean?

Answer 28

A zero matrix

Question 29

What are the
dimensions of the
zero matrix in the equation
B + O = B
given that:

Answer 29

2 x 3

If the dimensions of a zero matrix aren’t given, it’s understood that the dimensions match the dimensions of matrix B.

Question 30

How would you
notate a
zero matrix with
two rows and four columns?

Answer 30

O_2x4

A zero matrix is indicated by O,
and a subscript can be added to indicate the dimensions of the matrix if necessary.

Question 31

Given

matrices A and O,

A − O = _____?

Answer 31

A

When we add the
m x n zero matrix to
any m x n matrix A, we get
matrix A back.

Question 32

Given

matrix A,

A + −A = _____?

Answer 32

O

Question 33

What is the
commutative property of addition?

(applied to matrices)

Answer 33

A + B = B + A

You can
add two matrices in any order and
get the
same result.

Question 34

What is the
associative property of addition?

(applied to matrices)

Answer 34

(A + B) + C = A + (B + C)

You can
change the grouping in matrix addition and get the
same result.

For example, you can add matrix A to B first, and then add matrix C or you can add matrix B to C and then add this result to A.

Question 35

What is the
additive identity property?

(applied to matrices)

Answer 35

A + O = A

The
sum of
any matrix A and the
appropriate zero matrix is the
matrix A.

Question 36

What is the
additive inverse property?

(applied to matrices)

Answer 36

A + (−A) = O

The sum of a
real number and its opposite is
always 0, and so the
sum of any matrix and its opposite gives a
zero matrix.

Question 37

What is the
closure property of addition?

(applied to matrices)

Answer 37

• A* + B is a matrix of the
• *same dimensions** as
• A* and B.

Question 38

Due to what
property,
applied to the matrices below,
do we know that

A + B = B + A?

Answer 38

The
commutative property of addition

You can
add two matrices in any order and
get the
same result.

Question 39

Due to what
property,
applied to the matrices below,
do we know that

(A + B) + C = A + (B + C)?

Answer 39

The
associative property of addition

You can
change the grouping in matrix addition and get the
same result.

Question 40

Due to what
property,
applied to the matrices below,
do we know that

A + O = A?

Answer 40

The
additive identity property

The
sum of
any matrix A and the
appropriate zero matrix is the
matrix A.

Question 41

Due to what
property,
applied to the matrices below,
do we know that

A + (−A) = O?

Answer 41

What is the
additive inverse property?

The sum of a
real number and its opposite is
always 0, and so the
sum of any matrix and its opposite gives a
zero matrix.

Question 42

Due to what
property,
applied to the matrices below,
do we know that

• A* + B is a matrix of the
• *same dimensions** as
• A* and B.

Answer 42

The
closure property of addition

Question 43

What is the
associative property of multiplication?

(applied to matrices and scalars)

Answer 43

(cd)A = c(dA)

If a matrix is multiplied by
two scalars, you can
multiply the scalars together first,
and then
multiply by the matrix
OR
you can multiply the matrix by one scalar,
and then
the resulting matrix by the other.

Question 44

What are the
distributive properties?

(applied to matrices and scalars)

Answer 44

c(A + B) = c**A + c**B
and
(c + d)A = cA + dA

A
scalar can be
distributed over
matrix addition.

Question 45

What is the
multiplicative identity property?

(applied to matrices and scalars)

Answer 45

1A = A

Because 1 • a = a for any real number a, the scalar 1 will always be the multiplicative identity in scalar multiplication.

Question 46

What are the
multiplicative properties of zero?

(applied to matrices and scalars)

Answer 46

0 • A = 0
and
c • O = O

• In scalar multiplication, 0 times any m x n matrix A is the m x n zero matrix.*
• This is derived from the multiplicative properties of zero in the real number system.*

Question 47

What is the
closure property of multiplication?

(applied to matrices and scalars)

Answer 47

• cA* is a matrix of the
• *same dimensions** as
• A*.

Question 48

Due to what
property,
applied to the matrices and scalars below,
do we know that

(cd)A = c(dA)?

Answer 48

The
associative property of multiplication

If a matrix is multiplied by
two scalars, you can
multiply the scalars together first,
and then
multiply by the matrix
OR
you can multiply the matrix by one scalar,
and then
the resulting matrix by the other.

Question 49

Due to what
property,
applied to the matrices and scalars below,
do we know that

c(A + B) = c**A + c**B
and
(c + d)A = cA + dA?

Answer 49

The
distributive properties

A
scalar can be
distributed over
matrix addition.

Question 50

Due to what
property,
applied to the matrices and scalars below,
do we know that

1A = A?

Answer 50

The
multiplicative identity property

Because 1 • a = a for any real number a, the scalar 1 will always be the multiplicative identity in scalar multiplication.

Question 51

Due to what
property,
applied to the matrices and scalars below,
do we know that

0 • A = O
and
c • O = O

Answer 51

The
multiplicative properties of zero

• In scalar multiplication, 0 times any m x n matrix A is the m x n zero matrix.*
• This is derived from the multiplicative properties of zero in the real number system.*

Question 52

Due to what
property,
applied to the matrices and scalars below,
do we know that

• cA* is a matrix of the
• *same dimensions** as
• A*?

Answer 52

The
closure property of multiplication

Question 53

# _define_:
scalar multiplication

(applied to matrices)

Answer 53

The

• *product** of a
• *real number** and a
• *matrix**

Each entry in the matrix is
multiplied by the given
scalar.

Question 54

Answer 54

Question 55

# _define_:
*n*-tuple

Answer 55

An

• *ordered list** of
• n* numbers

Question 56

# _define_:
dot product

Answer 56

A

• *single number** obtained from
• *two n-tuples** by
• *summing the products** of the
• *respective entries**

Also called
scalar product

Question 57

How do you

• *notate** an
• n*-tuple using a
• *variable**?

Answer 57

By a

• *variable** with an
• *arrow on top**

Question 58

Answer 58

The
product of
two n-tuples of
equal length is
always a
single real number

Question 59

Answer 59

Take the

• *dot product** of the respective
• *rows of matrix A** and the
• *columns of matrix B**

Specifically, the entry
c_i,j is the
dot product of
_{<strong>→</strong> <strong>→</strong>}
a_i & b_j

For example . . .

Question 60

Generally,
how do you know whether
matrix multiplication is
defined?

Answer 60

The
number of columns in the first matrix
must be equal to the
number of rows in the second matrix

Question 61

Generally,
how do you know the
dimensions of the product of
matrix multiplication?

Answer 61

It will have
first matrix’s number of rows
and the
second matrix’s number of columns

(m x n) (n x k):
product is m x k

Question 62

Is this operation
defined?

If so,
what are the
dimensions of the
product?

Answer 62

Yes,
it’s defined

3 x 2
are the dimensions of the product

Question 63

Is this operation
defined?

If so,
what are the
dimensions of the
product?

Answer 63

No,
it’s undefined

The would-be factors are
3 x 4 and 3 x 2,
and because the
inside dimensions aren’t the same,
multiplication is undefined.

Question 64

# _define_:
identity matrix

Answer 64

A

• *square matrix** with
• *1’s along the diagonal** from the
• *upper left to bottom right** and
• *0’s everywhere else**

Question 65

Envision
the matrix
I₄.

Answer 65

The n x n identity matrix, denoted
I_n,
is a matrix with n rows and n columns.

Question 66

Answer 66

The
product of
any square matrix and the
appropriate identity matrix is always the
original matrix,
regardless of the order in which the
multiplication was performed

Question 67

Answer 67

The
product of
any square matrix and the
appropriate identity matrix is always the
original matrix,
regardless of the order in which the
multiplication was performed

Question 68

Applied to the matrices below,
which of the following
is not true?

Answer 68

Question 69

# _define_:
transformation

Answer 69

The
same thing as a
function
(something which
takes in a number and
outputs a number)

BUT while
functions are typically visualized with graphs,
“transformations” are typically visualized as some object moving, stretching, squashing, etc.

Question 70

How do you represent a
two-dimensional linear transform
with a
matrix?

Answer 70

These are the respective
x- and y-coordinates where the points
(1, 0) and (0, 1)
end up

Note the relationship with I₂

Question 71

How do

• *two dimensional linear transformations**
• *relate** to
• *I₂**?

Answer 71

These

• *transformation matrices** are
• *scaled** forms of
• *I₂**

Question 72

Given a
two-dimensional linear transformation,
how do you
determine where a
given vector
ended up?

Answer 72

(x, y) = the vector before transform

(a, c) = where (1, 0) ended

(b, d) = where (0, 1) ended)

Question 73

Given
matrix A,
what does
| A | mean?

Answer 73

The
determinant of matrix A

Question 74

Given
matrix A_2x2,
how do you
calculate the
determinant of A?

Answer 74

Question 75

What is the

• *adjugate** of
• *matrix A** below?

Answer 75

Values in the

• descending diagonal are swapped
• ascending diagonal are made opposite

Question 76

Given
matrix A,

___ • A = I

Answer 76

A⁻¹ • A = I

Also . . .

A • A⁻¹ = I

(Pronounced
“the inverse of matrix A”)

Question 77

Given
matrix A,
how do you determine
A⁻¹?

Answer 77

A⁻¹ = 1 • adj(A)
| A |

One over the determinant of A
times the
adjugate of A

Question 78

How do you
know whether a
matrix is
invertible?

Answer 78

A
matrix is invertible
unless the
determinant equals zero

This would require you to divide by zero when determining the inverse, which is an undefined operation

Question 79

Is
matrix A
invertible?

Answer 79

No

A matrix is invertible
unless its
determinant equals zero

Here . . .

A | = 2•6 − 4•3 = 0

Question 80

Is
matrix A
invertible?

Answer 80

Yes

A matrix is invertible
unless its
determinant equals zero

Here . . .

A | = 2•6 − 5•3 = −3

Question 81

What are the

• *steps** to
• *solving a linear system** with
• *matrix equations**?

i.e.

2s – 5t = 7
–2s + 4 = –6

Answer 81

1. Represent the system as a matrix
2. Multiply each side by the inverse of the matrix, which isolates the variables
3. Simplify