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Mathematics
Problem
Consider the matrices
P
=
[
1
2
0
−
1
1
3
]
P=\begin{bmatrix}1&2&0\\-1&1&3\end{bmatrix}
P
=
[
1
−
1
2
1
0
3
]
,
Q
=
[
2
1
0
1
1
−
2
]
Q=\begin{bmatrix}2&1\\0&1\\1&-2\end{bmatrix}
Q
=
2
0
1
1
1
−
2
and
R
=
[
3
−
1
2
0
]
R=\begin{bmatrix}3&-1\\2&0\end{bmatrix}
R
=
[
3
2
−
1
0
]
. a. (i) Find
Q
R
QR
QR
. b. (ii) Find
P
(
Q
R
)
P(QR)
P
(
QR
)
. c. (iii) Find
P
Q
PQ
P
Q
. d. (iv) Find
(
P
Q
)
R
(PQ)R
(
P
Q
)
R
. e. (v) Prove that
P
(
Q
R
)
=
(
P
Q
)
R
P(QR)=(PQ)R
P
(
QR
)
=
(
P
Q
)
R
for the given matrices.
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