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Invertible Matrix
Related Topics
Wize University Linear Algebra Textbook > Matrices
Matrix Inverse
4 Activities
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?
A
=
U
B
A=UB
A
=
U
B
A
=
U
−
1
B
A=U^{-1}B
A
=
U
−
1
B
B
=
U
−
1
A
B=U^{-1}A
B
=
U
−
1
A
B
=
A
U
−
1
B=AU^{-1}
B
=
A
U
−
1
None of the above
I don't know
Check Submission
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
?
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
.
Practice: Matrix Inverse
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
:
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
])
?