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Practice: Matrix Inverse
Related Topics
Wize University Linear Algebra Textbook > Matrices
Matrix Multiplication
6 Activities
Wize University Linear Algebra Textbook > Matrices
Matrix Inverse
4 Activities
Let
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
be size 10 by 10 invertible matrices: Select the expression which is equal to the matrix
(
A
(
B
C
)
−
1
D
)
T
\left(A\left(BC\right)^{-1}D\right)^T
(
A
(
B
C
)
−
1
D
)
T
:
I don't know
Check Submission
More Matrix Multiplication Questions:
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 25
Show that
A
‾
T
A
‾
\bcb{\boldsymbol{ \mtran{\A}\ul{A}}} \,
A
T
A
is a symmetric matrix for any
n
×
n
\bcb{\boldsymbol{ n \times n}}
n
×
n
matrix
A
‾
\bcb{\boldsymbol{ \ul{A} }}
A
.
19.4F_WML_6_$\tkco{eg3}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{3}$_
Recall that matrix-vector multiplication of the form
A
‾
x
⃗
=
b
⃗
\A \vx = \vb
A
x
=
b
can be thought of as taking the dot product of the rows of the matrix
A
‾
\A
A
with the column vector
x
⃗
\vx
x
. If
A
⃗
1
j
\vec{A}_{1j}
A
1
j
represents the first row of the matrix
A
‾
\A
A
, write the following matrix-vector multiplication as a system of vector-vector products:
[
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
]
[
x
1
x
2
x
3
]
=
[
b
1
b
2
b
3
]
\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} % \colvecth{x_1}{x_2}{x_3} % = % \colvecth{b_1}{b_2}{b_3}
a
11
a
21
a
31
a
12
a
22
a
32
a
13
a
23
a
33
x
1
x
2
x
3
=
b
1
b
2
b
3
133 - FML 3 - 18.1W e.g. 74
Find
A
‾
7
\bcb{\boldsymbol{ \ul{A}^7}}
A
7
if
A
‾
=
[
2
4
2
0
]
\bcb{\boldsymbol{ \ul{A } = \begin{bmatrix} 2 & 4 \\ 2 & 0 \end{bmatrix} }}
A
=
[
2
2
4
0
]
.
133 - FML 3 - 18.1W e.g. 5
Given the matrix
A
‾
=
[
a
b
c
d
]
\bcb{\boldsymbol{ \underline{A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} }}
A
=
[
a
c
b
d
]
, find
A
‾
⋅
ı
^
\bcb{\boldsymbol{ \underline{A} \cdot \ihat}}
A
⋅
^
. Repeat for
A
‾
⋅
ȷ
^
\bcb{\boldsymbol{ \underline{A}\cdot \jhat}}
A
⋅
^
. What do you notice?
A
‾
⋅
ı
^
=
\A \cdot \ihat = \,
A
⋅
^
=
a
c
A
‾
⋅
ȷ
^
=
\A \cdot \jhat = \,
A
⋅
^
=
b
d
If
A
=
[
2
1
−
2
5
]
A=\begin{bmatrix}2&1\\-2&5\end{bmatrix}
A
=
[
2
−
2
1
5
]
and
B
=
[
4
−
3
3
2
]
B=\begin{bmatrix}4&-3\\3&2\end{bmatrix}
B
=
[
4
3
−
3
2
]
, find (1,2)-entry of
(
2
A
−
1
(
3
B
)
T
)
−
1
\left(2A^{-1}\left(3B\right)^T\right)^{-1}
(
2
A
−
1
(
3
B
)
T
)
−
1
If
A
=
[
2
1
−
2
5
]
A = \begin{bmatrix} 2 & 1 \\ -2 & 5 \end{bmatrix}
A
=
[
2
−
2
1
5
]
and
B
=
[
4
−
3
3
2
]
B = \begin{bmatrix} 4 & -3 \\ 3 & 2 \end{bmatrix}
B
=
[
4
3
−
3
2
]
, find the (1,2) entry of
(
2
A
−
1
(
3
B
)
T
)
−
1
(2A^{-1}(3B)^T)^{-1}
(
2
A
−
1
(
3
B
)
T
)
−
1
Practice: Matrix Powers (~2014 Test2 #14)
Practice: Matrix Powers
Given the matrix
A
=
[
1
0
1
1
]
A=\begin{bmatrix} 1&0\\1&1 \end{bmatrix}
A
=
[
1
1
0
1
]
, find
A
n
A^n
A
n
for integer values
n
>
1
n>1
n
>
1
Matrix Multiplication
Practice: Matrix Multiplication
Compute
M
N
MN
M
N
and
N
M
NM
N
M
, where
M
=
[
6
−
1
0
3
]
,
N
=
[
−
1
−
3
0
1
]
M=\left[ \begin{array}{rrrr} 6&-1&0&3\end{array} \right] ,\qquad N=\left[ \begin{array}{r} -1\\[0.5em] -3\\[0.5em] 0\\[0.5em] 1 \end{array} \right]
M
=
[
6
−
1
0
3
]
,
N
=
−
1
−
3
0
1
Matrix Transpose
Given
C
=
[
4
3
1
]
C = \left[ \begin{array}{rrr} 4 & 3 & 1 \end{array} \right]
C
=
[
4
3
1
]
and
D
=
[
1
3
1
2
4
5
]
D = \left[ \begin{array}{rrr} 1 & 3 & 1\\ 2 & 4 & 5\\ \end{array} \right]
D
=
[
1
2
3
4
1
5
]
, find the following entries.
Practice Question 3: Matrix Multiplication
Practice Question: Matrix Multiplication
Given the matrices
A
=
[
1
−
1
0
2
3
−
3
]
A=\begin{bmatrix}1&-1\\0&2\\3&-3\end{bmatrix}
A
=
1
0
3
−
1
2
−
3
,
B
=
[
1
1
5
4
0
0
]
B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}
B
=
1
5
0
1
4
0
, and
C
=
[
0
0
0
0
1
−
4
]
C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}
C
=
[
0
0
0
1
0
−
4
]
, if
D
=
A
C
D=AC
D
=
A
C
, find
D
23
D_{23}
D
23
. (i.e. find the (2, 3)-entry of
D
D
D
).
Matrix Multiplication
Practice: Matrix Multiplication
A
=
[
−
3
3
1
−
2
2
0
−
1
1
2
]
,
B
=
[
2
−
1
−
3
0
1
−
1
0
0
3
]
A=\left[ \begin{array}{rrr} -3&3&1\\ -2&2&0\\ -1&1&2 \end{array} \right] ,\qquad B=\left[ \begin{array}{rrr} 2&-1&-3\\ 0&1&-1\\ 0&0&3 \end{array} \right]
A
=
−
3
−
2
−
1
3
2
1
1
0
2
,
B
=
2
0
0
−
1
1
0
−
3
−
1
3
Practice: Matrix Powers
Practice: Matrix Powers
Given the matrix
A
=
[
1
−
1
1
−
1
]
A=\begin{bmatrix}1&-1\\1&-1\end{bmatrix}
A
=
[
1
1
−
1
−
1
]
, find
A
n
A^n
A
n
, where
n
n
n
is an integer greater than 1.
Practice: Matrix Operations
Given the matrices below, which of the following statements is/are defined?
A
=
[
a
b
c
d
]
,
B
=
[
e
f
g
h
i
j
k
l
]
,
C
=
[
m
n
o
p
q
r
s
t
]
A=\left[\begin{array}{c} a&b\\c&d \end{array}\right], \ B=\left[\begin{array}{c} e&f&g&h\\i&j&k&l \end{array}\right],\ C=\left[\begin{array}{c} m&n\\o&p\\q&r\\s&t \end{array}\right]
A
=
[
a
c
b
d
]
,
B
=
[
e
i
f
j
g
k
h
l
]
,
C
=
m
o
q
s
n
p
r
t
Practice: Matrix Multiplication (~2016 Test2 #13)
Practice: Matrix Multiplication
Given the matrices
A
=
[
1
−
1
0
2
3
−
3
]
A=\begin{bmatrix}1&-1\\0&2\\3&-3\end{bmatrix}
A
=
1
0
3
−
1
2
−
3
,
B
=
[
1
1
5
4
0
0
]
B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}
B
=
1
5
0
1
4
0
, and
C
=
[
0
0
0
0
1
−
4
]
C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}
C
=
[
0
0
0
1
0
−
4
]
, if
D
=
A
C
D=AC
D
=
A
C
, find
D
23
D_{23}
D
23
. (i.e. find the (2, 3)-entry of
D
D
D
).
Matrix Multiplication
Practice: Matrix Multiplication
M
=
[
6
−
1
0
3
]
,
N
=
[
−
1
−
3
0
1
]
M=\left[ \begin{array}{rrrr} 6&-1&0&3\end{array} \right] ,\qquad N=\left[ \begin{array}{r} -1\\[0.5em] -3\\[0.5em] 0\\[0.5em] 1 \end{array} \right]
M
=
[
6
−
1
0
3
]
,
N
=
−
1
−
3
0
1
Practice Question 1: Matrix Operations
Practice Question: Matrix Operations
Given the matrices
A
=
[
1
−
1
0
2
3
−
3
]
A=\begin{bmatrix}1&-1\\0&2\\3&-3\end{bmatrix}
A
=
1
0
3
−
1
2
−
3
,
B
=
[
1
1
5
4
0
0
]
B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}
B
=
1
5
0
1
4
0
, and
C
=
[
0
0
0
0
1
−
4
]
C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}
C
=
[
0
0
0
1
0
−
4
]
, which of the following is/are defined?
i.)
(
B
C
)
T
(BC)^T
(
B
C
)
T
Practice: Matrix Operations (~2014 Test2 #9)
Practice: Matrix Operations
Given the matrices
A
=
[
1
−
1
0
2
3
−
3
]
A=\begin{bmatrix}1&-1\\0&2\\3&-3\end{bmatrix}
A
=
1
0
3
−
1
2
−
3
,
B
=
[
1
1
5
4
0
0
]
B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}
B
=
1
5
0
1
4
0
, and
C
=
[
0
0
0
0
1
−
4
]
C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}
C
=
[
0
0
0
1
0
−
4
]
, which of the following is/are defined? (select all that apply)
Practice: Matrix Operations (~2014 Test2 #12)
Practice: Matrix Operations
Given the matrix
B
=
[
2
0
1
0
1
3
0
0
2
]
B=\begin{bmatrix} 2&0&1\\0&1&3\\0&0&2 \end{bmatrix}
B
=
2
0
0
0
1
0
1
3
2
, find the third column of
(
B
−
2
I
3
)
2
(B-2I_3)^2
(
B
−
2
I
3
)
2
.
Matrix Properties With Powers
For each matrix A, find A
3
if it exists
a
)
A
=
[
2
2
2
2
]
b
)
A
=
[
2
2
2
2
2
2
]
a) A=\begin{bmatrix}2&2\\2&2\\\end{bmatrix} \space \space b) \space A=\begin{bmatrix}2&2&2\\2&2&2\\\end{bmatrix}
a
)
A
=
[
2
2
2
2
]
b
)
A
=
[
2
2
2
2
2
2
]
Practice Question: Matrix Multiplication Criteria
Practice Question: Properties of Matrices
𝐴 is a 2 × 5 matrix, 𝐵 is a 5 × 6 matrix, 𝐶 is a 6 × 2 matrix, and 𝐷 is a 1 × 2 row matrix. Which one of the following is defined?
a.
A
B
+
2
C
AB+2C
A
B
+
2
C
Matrix Multiplication
Prove or disprove that for all square matrices of the same size:
A
‾
2
−
B
‾
2
=
(
A
‾
+
B
‾
)
(
A
‾
−
B
‾
)
\underline{A}^2-\underline{B}^2=(\underline{A}+\underline{B})(\underline{A}-\underline{B})
A
2
−
B
2
=
(
A
+
B
)
(
A
−
B
)
Practice: Matrix Multiplication
Consider the matrices and vectors defined below. Which of the multiplications below are not well defined?
A
=
[
2
4
5
−
5
]
A=\begin{bmatrix}2 & 4\\5 & -5\end{bmatrix}
A
=
[
2
5
4
−
5
]
,
B
=
[
1
−
2
]
B=\begin{bmatrix}1 & -2\end{bmatrix}
B
=
[
1
−
2
]
and
C
=
[
0
2
]
C=\begin{bmatrix}0\\2\end{bmatrix}
C
=
[
0
2
]
Practice: Matrix Multiplication (~2015 Test2 #13)
Practice: Matrix Multiplication
Given the matrices
A
=
[
1
−
1
0
2
3
−
3
]
A=\begin{bmatrix}1&-1\\0&2\\3&-3\end{bmatrix}
A
=
1
0
3
−
1
2
−
3
,
B
=
[
1
1
5
4
0
0
]
B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}
B
=
1
5
0
1
4
0
, and
C
=
[
0
0
0
0
1
−
4
]
C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}
C
=
[
0
0
0
1
0
−
4
]
, find the second row of
(
B
C
)
T
(BC)^T
(
B
C
)
T
.
Practice: Matrix Multiplication of different sized matrices
Let
M
=
[
6
−
1
0
3
]
M=\left[ \begin{array}{rrrr} 6&-1&0&3\end{array} \right]
M
=
[
6
−
1
0
3
]
and
N
=
[
−
1
−
3
0
1
]
N=\left[ \begin{array}{r} -1\\[0.5em] -3\\[0.5em] 0\\[0.5em] 1 \end{array} \right]
N
=
−
1
−
3
0
1
Compute the products
M
N
MN
M
N
and
N
M
NM
N
M
Practice: Matrix Multiplication of 3x3 matrices
Let
A
=
[
−
3
3
1
−
2
2
0
−
1
1
2
]
A=\left[ \begin{array}{rrr} -3&3&1\\ -2&2&0\\ -1&1&2 \end{array} \right]
A
=
−
3
−
2
−
1
3
2
1
1
0
2
and
B
=
[
2
−
1
−
3
0
1
−
1
0
0
3
]
B=\left[ \begin{array}{rrr} 2&-1&-3\\ 0&1&-1\\ 0&0&3 \end{array} \right]
B
=
2
0
0
−
1
1
0
−
3
−
1
3
Compute the products
A
B
AB
A
B
,
B
A
BA
B
A
, and
A
2
A^2
A
2
Matrix Multiplication
e.g. Prove or disprove that for all square matrices of the same size,
A
‾
2
−
B
‾
2
=
(
A
‾
+
B
‾
)
(
A
‾
−
B
‾
)
\underline{A}^2-\underline{B}^2=(\underline{A}+\underline{B})(\underline{A}-\underline{B})
A
2
−
B
2
=
(
A
+
B
)
(
A
−
B
)
Matrix problem
Suppose that
A
=
[
1
/
5
2
0.88
−
152
0.03
11
1
−
1
0
]
A=\begin{bmatrix}1/5 & 2 & 0.88\\-152 & 0.03 & 11\\1 & -1 & 0\end{bmatrix}
A
=
1/5
−
152
1
2
0.03
−
1
0.88
11
0
and
B
=
[
15
2
−
15
13
0.38
−
16.78
]
B=\begin{bmatrix}15 & 2\\-15 & 13\\0.38 & -16.78\end{bmatrix}
B
=
15
−
15
0.38
2
13
−
16.78
. Find the (3,2)-entry of
A
B
AB
A
B
(i.e.
A
B
32
AB_{32}
A
B
32
).
Matrix Operations
Suppose that
A
A
A
and
B
B
B
are both
3
×
3
3\times3
3
×
3
matrices. Find a matrix
B
B
B
such that
B
A
BA
B
A
results in swapping row 1 and row 2 in
A
A
A
, followed by multiplying row 3 in
A
A
A
by 4.
Matrix Multiplication
Consider the matrices and vectors defined below. Which of the multiplications below are not well defined?
A
=
[
2
4
5
−
5
]
A=\begin{bmatrix}2 & 4\\5 & -5\end{bmatrix}
A
=
[
2
5
4
−
5
]
,
B
=
[
1
−
2
]
B=\begin{bmatrix}1 & -2\end{bmatrix}
B
=
[
1
−
2
]
and
C
=
[
0
2
]
C=\begin{bmatrix}0\\2\end{bmatrix}
C
=
[
0
2
]
Suppose that
A
A
A
and
B
B
B
are both
3
×
3
3\times3
3
×
3
matrices. Find a matrix
B
B
B
such that
B
A
BA
B
A
results in swapping row 1 and row 2 in
A
A
A
, followed by multiplying row 3 in
A
A
A
by 4.
If
A
A
A
is a
2
×
4
2\times4
2
×
4
matrix,
B
B
B
is a
2
×
2
2\times2
2
×
2
matrix, and
C
C
C
is a
4
×
2
4\times2
4
×
2
matrix, which of the following expressions is NOT defined?
Suppose that
A
=
[
1
/
5
2
0.88
−
152
0.03
11
1
−
1
0
]
A=\begin{bmatrix}1/5 & 2 & 0.88\\-152 & 0.03 & 11\\1 & -1 & 0\end{bmatrix}
A
=
1/5
−
152
1
2
0.03
−
1
0.88
11
0
and
B
=
[
15
2
−
15
13
0.38
−
16.78
]
B=\begin{bmatrix}15 & 2\\-15 & 13\\0.38 & -16.78\end{bmatrix}
B
=
15
−
15
0.38
2
13
−
16.78
Find the (3,2)-entry of
A
B
AB
A
B
(i.e.
A
B
32
AB_{32}
A
B
32
).
133 - FML 1 - 18.1W e.g. 8
If
B
‾
=
diag
(
β
1
,
β
2
)
\B=\text{diag}(\beta_1,\beta_2)
B
=
diag
(
β
1
,
β
2
)
find
B
‾
2
\B^2
B
2
.
Practice: Matrix Multiplication
Given that
A
=
[
1
3
−
1
0
2
1
]
[
−
2
2
0
0
1
−
1
4
3
−
1
]
A=\left[\begin{array}{c} 1&3&-1\\ 0&2&1 \end{array}\right] \left[\begin{array}{c} -2&2&0\\ 0&1&-1\\ 4&3&-1 \end{array}\right]
A
=
[
1
0
3
2
−
1
1
]
−
2
0
4
2
1
3
0
−
1
−
1
, find
A
21
A_{21}
A
21
(i.e. the (2,1)-entry of the matrix A).
Practice: Matrix Properties (~2014 Test2 #13)
Practice: Matrix Properties
Let
A
A
A
and
B
B
B
be
n
×
n
n\times n
n
×
n
matrices such that
A
B
=
O
AB=O
A
B
=
O
, where matrix
O
O
O
is the
n
×
n
n\times n
n
×
n
zero matrix.
Which of the following statement must always be true for any
n
n
n
value?
Given the two matrices below, find
A
B
AB
A
B
and
B
A
BA
B
A
.
A
=
[
4
−
4
7
−
6
1
4
0
−
4
0
]
,
B
=
[
5
−
4
−
3
−
1
5
−
4
3
−
1
5
]
A=\begin{bmatrix}4&-4&7\\-6&1&4\\0&-4&0\end{bmatrix}, \quad B=\begin{bmatrix}5&-4&-3\\-1&5&-4\\3&-1&5\end{bmatrix}
A
=
4
−
6
0
−
4
1
−
4
7
4
0
,
B
=
5
−
1
3
−
4
5
−
1
−
3
−
4
5
Practice Question: Matrix Multiplication
Practice Question: Matrix Multiplication
Given the matrices
B
=
[
1
1
5
4
0
0
]
B=\begin{bmatrix}1&1\\5&4\\0&0\end{bmatrix}
B
=
1
5
0
1
4
0
, and
C
=
[
0
0
0
0
1
−
4
]
C=\begin{bmatrix}0&0&0\\0&1&-4\end{bmatrix}
C
=
[
0
0
0
1
0
−
4
]
, find the second row of
(
B
C
)
T
(BC)^T
(
B
C
)
T
.
Matrix Multiplication
e.g. If
A
‾
=
[
−
8
10
−
3
−
7
]
\A=\begin{bmatrix} -8&10\\-3&-7 \end{bmatrix}
A
=
[
−
8
−
3
10
−
7
]
and
B
‾
=
[
−
2
0
4
6
]
\B=\begin{bmatrix} -2&0\\4&6 \end{bmatrix}\!
B
=
[
−
2
4
0
6
]
,
find
A
B
A B
A
B
Matrix Multiplication
If
A
‾
=
[
−
8
10
−
3
−
7
]
\bcb{ {\A}=\begin{bmatrix} -8&10\\-3&-7 \end{bmatrix}}
A
=
[
−
8
−
3
10
−
7
]
and
B
‾
=
[
−
2
0
4
6
]
\bcb{ \B=\begin{bmatrix} -2&0\\4&6 \end{bmatrix}\!}
B
=
[
−
2
4
0
6
]
, find
A
‾
B
‾
\bcb{\A \B}
A
B
.
Example: Matrix Multiplication
Example:
Computer the following matrix multiplications (matrix products):
a.)
[
1
−
2
4
]
[
0
3
−
3
]
\begin{bmatrix}1 &−2 &4\end{bmatrix}\begin{bmatrix}0\\3\\-3\end{bmatrix}
[
1
−
2
4
]
0
3
−
3
Let
A
=
[
1
3
1
2
]
A=\begin{bmatrix}1 & 3\\1&2\end{bmatrix}
A
=
[
1
1
3
2
]
. Find all
2
×
2
2\times 2
2
×
2
matrices
B
B
B
such that
A
B
=
0
AB=0
A
B
=
0
.
Matrix Multiplication
Prove or disprove the following statement: if
B
‾
\underline{B}
B
has a column of zeros, then so does
A
‾
B
‾
\!\underline{A}\ \underline{B}\!
A
B
, if this product is defined.
Matrix and Scalar Multiplication
Let
A
=
[
2
3
0
−
1
]
A=\left[\begin{array}{rr}2&3\\0&-1\end{array}\right]
A
=
[
2
0
3
−
1
]
, and
B
=
[
−
1
−
2
7
3
]
B=\left[\begin{array}{rr}-1&-2\\7&3\end{array}\right]
B
=
[
−
1
7
−
2
3
]
, compute
3
A
B
3AB
3
A
B
Practice: Matrix Operations
Let
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
be invertible matrices of size
10
×
10
10\times 10
10
×
10
Show that:
(
A
(
B
C
)
−
1
D
)
T
=
D
T
(
B
T
)
−
1
(
C
T
)
−
1
A
T
\left(A\left(BC\right)^{-1}D\right)^T=D^T\left(B^{T}\right)^{-1}\left(C^{T}\right)^{-1}A^T
(
A
(
B
C
)
−
1
D
)
T
=
D
T
(
B
T
)
−
1
(
C
T
)
−
1
A
T
:
Matrix Multiplication and Identity
Let
A
=
[
−
1
0
1
0
−
2
2
0
1
0
]
A=\left[\begin{array}{rrr} -1&0&1\\0&-2&2\\0&1&0 \end{array}\right]
A
=
−
1
0
0
0
−
2
1
1
2
0
, find
(
A
+
I
)
2
(A+I)^2
(
A
+
I
)
2
Matrix Multiplication
Suppose that
A
=
[
1
0
−
1
2
]
A=\begin{bmatrix} 1&0\\-1&2 \end{bmatrix}
A
=
[
1
−
1
0
2
]
and
B
=
[
3
−
1
−
2
0
]
B=\begin{bmatrix} 3&-1\\-2&0 \end{bmatrix}
B
=
[
3
−
2
−
1
0
]
. Find
A
B
2
AB^2
A
B
2
.
Matrix Multiplication
Suppose that
A
=
[
2
−
1
1
1
0
3
]
A=\begin{bmatrix} 2&-1\\ 1&1\\ 0&3 \end{bmatrix}
A
=
2
1
0
−
1
1
3
and
B
=
[
0
3
−
2
−
1
1
0
]
B=\begin{bmatrix} 0&3\\ -2&-1\\ 1&0 \end{bmatrix}
B
=
0
−
2
1
3
−
1
0
.
Find
B
3
B^3
B
3
.
Matrix Multiplication and Transpose
Suppose that
A
=
[
2
−
1
1
1
0
3
]
A=\begin{bmatrix} 2&-1\\ 1&1\\ 0&3 \end{bmatrix}
A
=
2
1
0
−
1
1
3
and
B
=
[
0
3
−
2
−
1
1
0
]
B=\begin{bmatrix} 0&3\\ -2&-1\\ 1&0 \end{bmatrix}
B
=
0
−
2
1
3
−
1
0
.
Find
2
A
T
B
−
(
B
T
A
)
T
−
A
T
I
B
2A^TB-(B^TA)^T-A^TIB
2
A
T
B
−
(
B
T
A
)
T
−
A
T
I
B
, where
I
I
I
is the 3x3 identity matrix.
More Matrix Inverse Questions:
Properties of Matrix Inverse
Suppose
A
,
B
A,B
A
,
B
are any 2
n
×
n
n\times n
n
×
n
matrices, and that
A
+
B
A+B
A
+
B
is invertible.
Show that
A
(
A
+
B
)
−
1
B
=
B
(
A
+
B
)
−
1
A
A(A+B)^{-1}B=B(A+B)^{-1}A
A
(
A
+
B
)
−
1
B
=
B
(
A
+
B
)
−
1
A
.
133 - FML 3 - 18.1W e.g. 75
A
‾
=
[
1
2
3
−
2
0
1
−
1
−
2
3
]
\bcb{\boldsymbol{ \ul{A} = \begin{bmatrix} 1 & 2 & 3 \\ -2 & 0 & 1 \\ -1 & -2 & 3 \end{bmatrix} }}
A
=
1
−
2
−
1
2
0
−
2
3
1
3
then the
(
3
,
2
)
\bcb{\boldsymbol{ (3, 2)}}
(
3
,
2
)
entry in
adj
(
A
)
\bcb{\boldsymbol{ \text{adj}(A)}}
adj
(
A
)
is:
(a)
−
4
(b)
0
(c)
4
(d)
None
of
the
above
\text{(a)}\, \boldsymbol{ -4} \qquad\qquad\qquad \text{(b)}\; \boldsymbol{ 0} \qquad\qquad\qquad \text{(c)} \;\boldsymbol{ 4} \qquad\qquad\qquad \text{(d)} \;\text{\bf{None of the above}} \qquad\qquad\qquad
(a)
−
4
(b)
0
(c)
4
(d)
None of the above
Transpose of a Matrix
Prove that, for an invertible matrix
A
‾
\bcb{\A}
A
,
A
‾
T
\bcb{\mtran{\A}}
A
T
is also invertible, and
(
A
T
)
−
1
=
(
A
−
1
)
T
\bcb{(A^T )^ {−1} = (A^{−1} )^ T}
(
A
T
)
−
1
=
(
A
−
1
)
T
.
Given the matrix 𝐴
=
[
2
1
3
0
1
0
1
−
1
2
]
.
\text{Given the matrix 𝐴}=\begin{bmatrix} 2&1&3\\ 0&1&0\\ 1&-1&2\\ \end{bmatrix}.
Given the matrix
A
=
2
0
1
1
1
−
1
3
0
2
.
a) find the inverse of 𝐴.
b) solve the system of linear equations
2
𝑥
+
𝑦
+
3
𝑧
=
3
2𝑥+𝑦+3𝑧=3
2
x
+
y
+
3
z
=
3
Practice Question: Matrix Properties
Practice Question: Matrix Properties
Let
A
A
A
and
B
B
B
be invertible matrices. Which of the following statements is/are always true?
i.)
A
B
=
B
A
AB=BA
A
B
=
B
A
Practice Question: Matrix Inverse
Practice Question: Matrix Inverse
If
A
=
[
a
b
c
x
y
z
1
0
−
1
]
A=\left[\begin{array}{l} a&b&c\\ x&y&z\\ 1&0&-1 \end{array}\right]
A
=
a
x
1
b
y
0
c
z
−
1
is invertible, what is the third row of
A
A
−
1
AA^{-1}
A
A
−
1
?
Equation problem
Given that
A
2
=
(
I
−
2
A
)
(
I
+
A
)
A^2=(I-2A)(I+A)
A
2
=
(
I
−
2
A
)
(
I
+
A
)
and
A
A
A
is invertible, which of the following is an expression for
A
−
1
A^{-1}
A
−
1
?
Invertible Matrix
Let
A
A
A
and
B
B
B
be square matrices and
U
U
U
is an invertible matrix. Given that
B
U
=
A
BU=A
B
U
=
A
, which of the following statements is always true?
Given that
A
2
=
(
I
−
2
A
)
(
I
+
A
)
A^2=(I-2A)(I+A)
A
2
=
(
I
−
2
A
)
(
I
+
A
)
and
A
A
A
is invertible, which of the following is an expression for
A
−
1
A^{-1}
A
−
1
?
Let
A
A
A
and
B
B
B
be square matrices and
U
U
U
is an invertible matrix. Given that
B
U
=
A
BU=A
B
U
=
A
, which of the following statements is always true?
Let
A
A
A
and
B
B
B
be
n
×
n
n\times n
n
×
n
matrices, and let
k
k
k
be a constant. Which of the following statements is always true?
Suppose that
A
=
[
1
−
1
0
3
−
2
0
0
4
1
]
A=\begin{bmatrix}1 & -1 & 0\\3 & -2 & 0\\0 & 4 & 1\end{bmatrix}
A
=
1
3
0
−
1
−
2
4
0
0
1
and
B
B
B
is a
3
×
2
3\times2
3
×
2
matrix. If the product
A
B
=
[
3
4
0
1
1
−
2
]
AB = \begin{bmatrix}3 & 4\\0 & 1\\1 & -2\end{bmatrix}
A
B
=
3
0
1
4
1
−
2
, find the (1,2)-entry of
B
B
B
(i.e.
B
12
B_{12}
B
12
).
Practice: Matrix Inverse Application
Practice: Matrix Inverse Application
Let
A
=
[
1
2
0
−
1
]
A=\left[\begin{array}{c} 1&2\\0&-1 \end{array}\right]
A
=
[
1
0
2
−
1
]
and suppose that
3
A
B
−
[
1
−
1
2
0
]
=
I
2
3AB- \left[\begin{array}{c} 1&-1\\ 2&0 \end{array}\right] =I_2
3
A
B
−
[
1
2
−
1
0
]
=
I
2
.
Find the matrix
B
B
B
.
Practice: Matrix Properties (~2015 Test2 #16)
Practice: Matrix Properties
Let
A
A
A
and
B
B
B
be invertible matrices. Which of the following statements is/are always true? (select all that apply)
i.)
A
B
=
B
A
AB=BA
A
B
=
B
A
Inverse of a Matrix
e.g. Prove that, for an invertible matrix and a constant
c
c
c
,
c
A
‾
c\underline{ A}
c
A
is also invertible, and
(
c
A
‾
)
−
1
=
c
−
1
A
‾
−
1
\!(c \underline{A})^{ −1} = c^{ −1}\underline{A}^{−1}\!
(
c
A
)
−
1
=
c
−
1
A
−
1
.
Transpose of a Matrix
e.g. Prove that, for an invertible matrix
A
,
A
T
A, A^T
A
,
A
T
is also invertible, and
(
A
T
)
−
1
=
(
A
−
1
)
T
(A^T )^ {−1} = (A^{−1} )^ T
(
A
T
)
−
1
=
(
A
−
1
)
T
Let
A
A
A
and
B
B
B
be
n
×
n
n\times n
n
×
n
matrices, and let
k
k
k
be a constant. Select the statement that is always true.
Given that
A
=
[
3
−
3
2
−
2
4
6
1
−
1
0
]
A=\begin{bmatrix}3 & -3 & 2\\-2 & 4 & 6\\1 & -1 & 0\end{bmatrix}
A
=
3
−
2
1
−
3
4
−
1
2
6
0
, find the (3,3)-entry of
A
−
1
A^{-1}
A
−
1
if it exists.
Practice: Matrix Inverse (~2016 Test2 #17)
Practice: Matrix Inverse
If
[
1
a
x
2
b
y
0
c
z
]
\left[\begin{array}{c} 1&a&x\\ 2&b&y\\ 0&c&z \end{array}\right]
1
2
0
a
b
c
x
y
z
is the inverse of the matrix
[
1
p
0
−
3
q
4
2
r
1
]
\left[\begin{array}{c} 1&p&0\\ -3&q&4\\ 2&r&1 \end{array}\right]
1
−
3
2
p
q
r
0
4
1
, find the value of
r
r
r
.
Matrix Multiplication
Express the following matrix
A
‾
\bcb{ \A }
A
as a product of five (or fewer) elementary matrices); repeat for
A
‾
−
1
\bcb{ \minv{\A} }
A
−
1
.
A
‾
=
[
−
3
0
0
2
0
1
0
1
0
]
\bcb{ \A=\begin{bmatrix} -3&0&0\\2&0&1\\0&1&0 \end{bmatrix} }
A
=
−
3
2
0
0
0
1
0
1
0
Practice: Matrix Inverse (~2015 Test2 #17)
Practice: Matrix Inverse
If
A
=
[
a
b
c
x
y
z
1
0
−
1
]
A=\left[\begin{array}{l} a&b&c\\ x&y&z\\ 1&0&-1 \end{array}\right]
A
=
a
x
1
b
y
0
c
z
−
1
is invertible, what is the third row of
A
A
−
1
AA^{-1}
A
A
−
1
?
Practice: Matrix Inverse
If
A
=
[
a
b
c
d
e
f
g
h
i
]
A= \begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}
A
=
a
d
g
b
e
h
c
f
i
is an invertible matrix and
C
=
A
A
−
1
C=AA^{-1}
C
=
A
A
−
1
, find the first row of
C
C
C
.
Practice: Matrix Inverse
Let
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
be invertible matrices of size
10
×
10
10\times10
10
×
10
Select the expression which is equivalent to the matrix:
(
A
(
B
C
)
−
1
D
)
T
\left(A\left(BC\right)^{-1}D\right)^T
(
A
(
B
C
)
−
1
D
)
T
Practice: Matrix Inverse
Practice Question: Matrix Inverse
Let
A
=
[
1
3
1
2
]
A=\begin{bmatrix} 1&3\\ 1&2 \end{bmatrix}
A
=
[
1
1
3
2
]
. Find all
2
×
2
2\times2
2
×
2
matrices
B
B
B
such that
A
B
=
[
5
4
−
2
0
]
.
AB=\begin {bmatrix}5&4\\-2&0\end{bmatrix}.
A
B
=
[
5
−
2
4
0
]
.
Practice: Matrix Operations
Let
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
be invertible matrices of size
10
×
10
10\times 10
10
×
10
Show that:
(
A
(
B
C
)
−
1
D
)
T
=
D
T
(
B
T
)
−
1
(
C
T
)
−
1
A
T
\left(A\left(BC\right)^{-1}D\right)^T=D^T\left(B^{T}\right)^{-1}\left(C^{T}\right)^{-1}A^T
(
A
(
B
C
)
−
1
D
)
T
=
D
T
(
B
T
)
−
1
(
C
T
)
−
1
A
T
:
Matrix Multiplication
e.g. Express the following matrix
A
‾
\underline{A}
A
as a product of five (or fewer) elementary matrices); repeat for
A
‾
−
1
\!\underline{A}^{−1}\!
A
−
1
,
Invertible Matrix Theorem
Example: Invertible Matrix Theorem
Consider the linear system
𝐴
𝑥
⃗
=
𝑏
⃗
𝐴𝑥⃗=\vec{𝑏}
A
x
⃗
=
b
where
A
5
×
5
A_{5 \times 5}
A
5
×
5
is an
invertible
matrix. What is
r
a
n
k
(
[
𝐴
∣
b
⃗
]
)
{\rm rank}([𝐴\ |\ \vec{b}])
rank
([
A
∣
b
])
?