1
Q
# define: matrix
A
A
- *rectangular arrangement** of
- *numbers** into
- *rows and columns**.
2
Q
In plainspeak,
what is a
matrix?
A
A
compact representation of numbers.
3
Q
What are the
dimensions
of matrix B?
A
3 x 2
Pronounced “three by two.”
Rows x Columns
4
Q
- *Envision** a
- *2 x 3 matrix**.
A
Rows x Columns
5
Q
What is
- *another word** for a
- *matrix entry**?
A
Matrix element
6
Q
How could you
- *identify** the
- *entry −7** in
- *matrix G**?
A
g1,3
It’s the entry in the first row and the third column.
7
Q
What is
element g2,1?
A
18
It’s in the second row and the first column.
8
Q
# _define_: augmented matrix
A
A matrix that
- *represents** a
- *system of equations**.
9
Q
In an
augmented matrix,
what does each
row represent?
A
One equation
in the system of equations
10
Q
In an
augmented matrix,
what does each
column represent?
A
A
variable
or the
constant terms
in the system of equations
11
Q
- *Envision** the
- *augmented matrix** that
- *represents** the
- *system below**.
A
12
Q
- *Envision** the
- *augmented matrix** that
- *represents** the
- *system below**.
A
13
Q
- *Envision** the
- *augmented matrix** that
- *represents** the
- *system below**.
A
14
Q
What are the
- *three** elementary
- *matrix row operations**?
A
Switch any two rows
Add one row to another
Multiply a row by a
nonzero constant
15
Q
Why can you
switch any two rows in an
augmented matrix?
A
The
- *order** of the equations
- *doesn’t matter**.
16
Q
Why can you
add one row to another in an
augmented matrix?
A
Because you can
- *add two equal quantities** to
- *both sides** of an equation.
17
Q
Why can you
multiply a row by a nonzero constant in an
augmented matrix?
A
Because you can
- *multiply both sides** of an equation by the
- *same nonzero constant**.
18
Q
How do you
- *notate**
- *interchanging rows 1 and 2**?
A
19
Q
How do you
- *notate**
- *multiplying row 2 by three**?
A
20
Q
How do you
- *notate**
- *replacing row 2** with the
- *sum of rows 1 and 2**?
A
21
Q
A
Just add the corresponding entries.
22
Q
A
Just subtract the corresponding entries.
23
Q
A
Undefined
Cannot add or subtract matrices with different dimensions.
24
Q
When working with
matrices,
how do you
refer to
real numbers?
A
Scalars
Any real number that is
not a part of the matrix is a
scalar.
25
```
# _define_:
matrix equation
```
An
* *equation** in which the
* *variable** stands for a
* *matrix**.
26
27
```
# _define_:
zero matrix
```
A
* *matrix** in which
* *all entries are 0**.
28
When working with
**matrices**,
what does
***O* mean**?
**A zero matrix**
29
What are the
**dimensions** of the
**zero matrix** in the equation
***B* + *O* = *B***
given that:
**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.*
30
How would you
notate a
**zero matrix** with
**two rows and four columns**?
***O*2x4**
*A zero matrix is indicated by O,
and a subscript can be added to indicate the dimensions of the matrix if necessary.*
31
Given
**matrices *A* and *O***,
***A* − *O* = \_\_\_\_\_**?
***A***
*When we add the
m x n zero matrix to
any m x n matrix A, we get
matrix A back.*
32
Given
**matrix *A***,
***A* + −*A* = \_\_\_\_\_**?
***O***
33
What is the
**commutative property of addition**?
(applied to matrices)
***A* + *B* = *B* + *A***
*You can
_add two matrices in any order_ and
get the
_same result_.*
34
What is the
**associative property of addition**?
(applied to matrices)
**(*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.*
35
What is the
**additive identity property**?
| (applied to matrices)
***A* + *O* = *A***
*The
_sum_ of
_any matrix A_ and the
_appropriate zero matrix_ is the
_matrix A_.*
36
What is the
**additive inverse property**?
| (applied to matrices)
***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_.
37
What is the
**closure property of addition**?
(applied to matrices)
* **A* + *B*** is a matrix of the
* *same dimensions** as
* **A* and *B***.
38
Due to what
**property**,
_applied to the matrices_ below,
do we know that
***A* + *B* = *B* +** ***A**?*
The
**commutative property of addition**
## Footnote
*You can
_add two matrices in any order_ and
get the
_same result_.*
39
Due to what
**property**,
_applied to the matrices_ below,
do we know that
**(*A* + *B*) + *C* = *A* + (*B* + *C*)**?
The
**associative property of addition**
## Footnote
*You can
_change the grouping_ in matrix addition and get the
_same result_.*
40
Due to what
**property**,
_applied to the matrices_ below,
do we know that
***A* + *O* =** ***A**?*
The
**additive identity property**
## Footnote
*The
_sum_ of
_any matrix A_ and the
_appropriate zero matrix_ is the
_matrix A_.*
41
Due to what
**property**,
_applied to the matrices_ below,
do we know that
***A* + (*−A*) =** ***O***?
What is the
**additive inverse property**?
## Footnote
*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_.*
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***.
The
**closure property of addition**
43
What is the
**associative property of multiplication**?
(applied to matrices and scalars)
**(*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.*
44
What are the
**distributive properties**?
| (applied to matrices and scalars)
***c*(*A* + *B*) = *c**A* + *c**B***
and
**(*c* + *d*)*A* = *cA* + *dA***
## Footnote
*A
_scalar_ can be
_distributed_ over
_matrix addition_.*
45
What is the
**multiplicative identity property**?
(applied to matrices and scalars)
**1*A* = *A***
*Because 1 • a = a for any real number a, the scalar 1 will always be the multiplicative identity in scalar multiplication.*
46
What are the
**multiplicative properties of zero**?
(applied to matrices and scalars)
**0 • *A* = 0**
and
**c • *O* = *O***
## Footnote
* 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.*
47
What is the
**closure property of multiplication**?
(applied to matrices and scalars)
* **cA*** is a matrix of the
* *same dimensions** as
* **A***.
48
Due to what
**property**,
_applied to the matrices and scalars_ below,
do we know that
**(*cd*)*A* = c(*dA)***?
The
**associative property of multiplication**
## Footnote
*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.*
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***?
The
**distributive properties**
## Footnote
*A
_scalar_ can be
_distributed_ over
_matrix addition_.*
50
Due to what
**property**,
_applied to the matrices and scalars_ below,
do we know that
**1*A*** **= *A***?
The
**multiplicative identity property**
## Footnote
*Because 1 • a = a for any real number a, the scalar 1 will always be the multiplicative identity in scalar multiplication.*
51
Due to what
**property**,
_applied to the matrices and scalars_ below,
do we know that
**0 • *A* = *O***
and
***c* • *O* = *O***
The
**multiplicative properties of zero**
## Footnote
* 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.*
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***?
The
**closure property of multiplication**
53
```
# _define_:
scalar multiplication
```
(applied to matrices)
The
* *product** of a
* *real number** and a
* *matrix**
## Footnote
*_Each entry_ in the matrix is
_multiplied_ by the given
_scalar_.*
54
55
```
# _define_:
*n*-tuple
```
An
* *ordered list** of
* **n* numbers**
56
```
# _define_:
dot product
```
A
* *single number** obtained from
* *two *n*-tuples** by
* *summing the products** of the
* *respective entries**
## Footnote
*Also called
_scalar product_*
57
How do you
* *notate** an
* **n*-tuple** using a
* *variable**?
By a
* *variable** with an
* *arrow on top**
58
*The
_product_ of
_two n-tuples_ of
_equal length_ is
always a
_single real number_*
59
Take the
* *dot product** of the respective
* *rows of matrix *A*** and the
* *columns of matrix *B***
Specifically, the entry
**ci,j** is the
**dot product** of
→ →
**ai** & **bj**
*For example . . .*
60
_Generally_,
how do you know whether
**matrix multiplication** is
**defined**?
The
**number of columns in the first matrix**
must be equal to the
**number of rows in the second matrix**
61
_Generally_,
how do you know the
**dimensions of the product** of
**matrix multiplication**?
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*
62
Is this operation
**defined**?
If so,
what are the
**dimensions** of the
**product**?
**Yes**,
it's defined
**3 x 2**
are the dimensions of the product
63
Is this operation
**defined**?
If so,
what are the
**dimensions** of the
**product**?
**No**,
it's undefined
## Footnote
*The would-be factors are
3 x _4_ and _3_ x 2,
and because the
inside dimensions aren't the same,
multiplication is undefined.*
64
```
# _define_:
identity matrix
```
A
* *square matrix** with
* *1's along the diagonal** from the
* *upper left to bottom right** and
* *0's everywhere else**
65
**Envision**
the matrix
***I*4**.
*The n x n identity matrix, denoted
In,
is a matrix with n rows and n columns.*
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_
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_
68
Applied to the matrices below,
which of the following
**is not true**?
69
```
# _define_:
transformation
```
The
**same** thing as a
**function**
(something which
_takes in a number_ and
_outputs a number_)
## Footnote
*BUT while
functions are typically visualized with graphs,
_"transformations" are typically visualized as some object_ moving, stretching, squashing, etc.*
70
How do you represent a
**two-dimensional linear transform**
with a
**matrix**?
*These are the respective
x- and y-coordinates where the points*
(1, 0) *and* (0, 1)
*end up*
## Footnote
*Note the relationship with I2*
71
How do
* *two dimensional linear transformations**
* *relate** to
* *I2**?
These
* *transformation matrices** are
* *scaled** forms of
* *I2**
72
Given a
_two-dimensional linear transformation_,
how do you
**determine** where a
**given vector**
**ended up**?
(x, y) = the vector before transform
(a, c) = where (1, 0) ended
(b, d) = where (0, 1) ended)
73
Given
_matrix A_,
what does
**| A |** mean?
The
**determinant of matrix A**
74
Given
_matrix A2x2_,
how do you
calculate the
**determinant of A**?
75
What is the
* *adjugate** of
* *matrix A** below?
*Values in the*
* *_descending diagonal_ are _swapped_*
* *_ascending diagonal_ are made _opposite_*
76
Given
_matrix A_,
## Footnote
**\_\_\_ • A = I**
**A−1 • A = I**
*Also . . .*
**A • A−1 = I**
*(Pronounced
"the inverse of matrix A")*
77
Given
_matrix A_,
how do you determine
**A−1**?
**A−1 = _1_ • adj(A)
| A |**
*_One over the determinant of A_
_times_ the
_adjugate of A_*
78
How do you
know whether a
**matrix** is
**invertible**?
A
_matrix is invertible_
**unless** the
**determinant equals zero**
## Footnote
*This would require you to divide by zero when determining the inverse, which is an undefined operation*
79
Is
matrix A
**invertible**?
**No**
*A matrix is invertible
unless its
determinant equals zero*
*Here . . .*
| A | = 2•6 − 4•3 = 0
80
Is
matrix A
**invertible**?
**Yes**
*A matrix is invertible
unless its
determinant equals zero*
*Here . . .*
| A | = 2•6 − 5•3 = −3
81
What are the
* *steps** to
* *solving a linear system** with
* *matrix equations**?
*i.e.*
2s – 5t = 7
–2s + 4 = –6
1. Represent the system as a matrix
2. Multiply each side by the inverse of the matrix, which isolates the variables
3. Simplify
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