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Matrix Transpositions
bencejdanko
What is the transpose of the row vector
v
=
[
1
2
3
]
v = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}
v
=
[
1
2
3
]
?
[
−
1
−
2
−
3
]
\begin{bmatrix} -1 & -2 & -3 \end{bmatrix}
[
−
1
−
2
−
3
]
[
3
2
1
]
\begin{bmatrix} 3 & 2 & 1 \end{bmatrix}
[
3
2
1
]
[
1
2
3
]
\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}
1
2
3
If
A
i
j
A_{ij}
A
ij
represents the element in row
i
i
i
and column
j
j
j
of matrix
A
A
A
, what is the element at
(
i
,
j
)
(i, j)
(
i
,
j
)
of
A
T
A^T
A
T
?
A
j
i
A_{ji}
A
ji
1
/
A
i
j
1/A_{ij}
1/
A
ij
A
i
j
A_{ij}
A
ij
−
A
j
i
-A_{ji}
−
A
ji
What is the transpose of the product
(
A
B
)
T
(AB)^T
(
A
B
)
T
?
B
T
A
T
B^T A^T
B
T
A
T
A
B
T
AB^T
A
B
T
A
T
B
T
A^T B^T
A
T
B
T
A
T
B
A^T B
A
T
B
If a matrix
A
A
A
is equal to its transpose (
A
=
A
T
A = A^T
A
=
A
T
), the matrix is called:
Skew-symmetric
Identity
Symmetric
Inverse
Find
A
T
A^T
A
T
if:
A
=
[
1
2
3
4
]
A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
A
=
[
1
3
2
4
]
[
1
2
3
4
]
\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
[
1
3
2
4
]
[
4
3
2
1
]
\begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix}
[
4
2
3
1
]
[
1
3
2
4
]
\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}
[
1
2
3
4
]
What is
(
A
T
)
T
(A^T)^T
(
A
T
)
T
?
A
2
A^2
A
2
A
−
1
A^{-1}
A
−
1
A
A
A
I
I
I
Which of these matrices is its own transpose?
[
0
1
0
0
]
\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}
[
0
0
1
0
]
[
2
0
0
2
]
\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}
[
2
0
0
2
]
[
1
2
3
4
]
\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
[
1
3
2
4
]
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