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Practice: Matrix Operations
Related Topics
Wize University Linear Algebra Textbook > Matrices
Basics of Matrices
4 Activities
Wize University Linear Algebra Textbook > Matrices
Matrix Transpose
5 Activities
Given that
A
=
[
−
1
2
3
−
4
]
A=\begin{bmatrix} -1&2\\3&-4 \end{bmatrix}
A
=
[
−
1
3
2
−
4
]
and
B
=
[
1
1
−
2
2
]
B=\begin{bmatrix}1&1\\-2&2\end{bmatrix}
B
=
[
1
−
2
1
2
]
, let
C
=
(
3
A
−
2
B
T
+
I
2
)
T
C=(3A-2B^T+I_2)^T
C
=
(
3
A
−
2
B
T
+
I
2
)
T
. Determine
C
22
C_{22}
C
22
.
Answer
I don't know
Check Submission
More Basics of Matrices Questions:
Find
x
,
y
,
z
x,y,z
x
,
y
,
z
if
[
x
1
z
2
−
1
3
]
+
[
2
3
4
1
y
5
]
=
[
−
3
4
1
3
−
2
8
]
.
\begin{bmatrix} x&1&z\\2&-1&3 \end{bmatrix} + \begin{bmatrix} 2&3&4\\ 1&y&5 \end{bmatrix} = \begin{bmatrix} -3&4&1\\ 3&-2&8 \end{bmatrix}.
[
x
2
1
−
1
z
3
]
+
[
2
1
3
y
4
5
]
=
[
−
3
3
4
−
2
1
8
]
.
x
=
x=
x
=
-5
Example: Matrix Operations
Example:
Given the matrices
𝐴
=
[
1
0
0
−
3
−
2
2
]
𝐴=\begin{bmatrix}1&0\\0&-3\\-2&2\end{bmatrix}
A
=
1
0
−
2
0
−
3
2
,
B
=
[
3
0
2
0
1
−
2
]
\text{B}=\begin{bmatrix}3&0\\2&0\\1&-2\end{bmatrix}
B
=
3
2
1
0
0
−
2
, and
C
=
[
1
2
0
1
0
−
1
]
C=\begin{bmatrix}1&2&0\\1&0&-1\end{bmatrix}
C
=
[
1
1
2
0
0
−
1
]
,
a) Find
2
A
−
B
2A-B
2
A
−
B
.
Matrix Basics
Given the matrix
A
=
[
2
0
1
0
1
3
0
0
2
]
A=\begin{bmatrix} 2&0&1\\0&1&3\\0&0&2 \end{bmatrix}
A
=
2
0
0
0
1
0
1
3
2
, define
B
=
A
+
0
3
×
3
−
2
I
3
B=A+0_{3\times3}-2I_3
B
=
A
+
0
3
×
3
−
2
I
3
and find the following entries.
Basics of Matrices
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
A
B-3A
B
−
3
A
.
More Matrix Transpose 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
.
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
.
Practice Question 8: Matrix Transpose
Practice Question: Matrix Transpose
Given the following matrices, find the values of
a
,
b
a, b
a
,
b
and
c
c
c
such that
−
3
A
=
B
T
-3A=B^T
−
3
A
=
B
T
.
A
=
[
1
3
−
6
0
15
2
]
A=\begin{bmatrix} 1&3&-6\\0&15&2 \end{bmatrix}
A
=
[
1
0
3
15
−
6
2
]
and
B
=
[
−
3
a
−
9
5
b
2
c
−
6
]
B=\begin{bmatrix}-3&a\\-9&5b\\2c&-6\end{bmatrix}
B
=
−
3
−
9
2
c
a
5
b
−
6
Matrix Transpose
Prove that for any square matrix
A
A
A
, the matrix
A
−
A
T
A-A^T
A
−
A
T
is
skew-symmetric
.
Example: Matrix Operations
Example:
Given the matrices
𝐴
=
[
1
0
0
−
3
−
2
2
]
𝐴=\begin{bmatrix}1&0\\0&-3\\-2&2\end{bmatrix}
A
=
1
0
−
2
0
−
3
2
,
B
=
[
3
0
2
0
1
−
2
]
\text{B}=\begin{bmatrix}3&0\\2&0\\1&-2\end{bmatrix}
B
=
3
2
1
0
0
−
2
, and
C
=
[
1
2
0
1
0
−
1
]
C=\begin{bmatrix}1&2&0\\1&0&-1\end{bmatrix}
C
=
[
1
1
2
0
0
−
1
]
,
a) Find
2
A
−
B
2A-B
2
A
−
B
.
Example: Matrix Transpose
Example:
Find the transpose of
𝐴
=
[
1
−
3
−
2
4
0
5
]
𝐴=\begin{bmatrix}1&-3\\-2&4\\0&5\end{bmatrix}
A
=
1
−
2
0
−
3
4
5
, then state the sizes of
A
A
A
and
A
T
A^T
A
T
.
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.