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Joint Entrance Examination
Mathematics
Matrices and Determinants
Question 316
Question 316 of 472
Medium
MCQ
Let \[A = \left( {
(
1
2
3
4
)
\begin{pmatrix} 1 & 2 \\ 3 & 4 \\ \end{pmatrix}
(
1
3
2
4
)
} \right)\] and \[B = \left( {
(
a
0
0
b
)
\begin{pmatrix} a & 0 \\ 0 & b \\ \end{pmatrix}
(
a
0
0
b
)
} \right),a,b \in N.\] Then
A
there cannot exist any \[B\] such that \[AB=BA\]
B
there exist more then one but finite number of \[B'\]s such that \[AB=BA\]
C
there exists exactly one \[B\] such that \[AB=BA\]
D
there exist infinitely many \[B'\]s such that \[AB=BA\]
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Official solution
\[A = \left[ {
(
1
2
3
4
)
\begin{pmatrix} 1 & 2 \\ 3 & 4 \\ \end{pmatrix}
(
1
3
2
4
)
} \right]\,\,\,\,B = \left[ {
(
a
0
0
b
)
\begin{pmatrix} a & 0 \\ 0 & b \\ \end{pmatrix}
(
a
0
0
b
)
} \right]\] \[AB = \left[ {
(
a
2
b
3
a
4
b
)
\begin{pmatrix} a & {2b} \\ {3a} & {4b} \\ \end{pmatrix}
(
a
3
a
2
b
4
b
)
} \right]\] \[BA = \left[ {
(
a
0
0
b
)
\begin{pmatrix} a & 0 \\ 0 & b \\ \end{pmatrix}
(
a
0
0
b
)
} \right]\left[ {
(
1
2
3
4
)
\begin{pmatrix} 1 & 2 \\ 3 & 4 \\ \end{pmatrix}
(
1
3
2
4
)
} \right] = \left[ {
(
a
2
a
3
b
4
b
)
\begin{pmatrix} a & {2a} \\ {3b} & {4b} \\ \end{pmatrix}
(
a
3
b
2
a
4
b
)
} \right]\] Hence,
A
B
=
B
A
AB=BA
A
B
=
B
A
only when
a
=
b
a=b
a
=
b
\[\therefore\] There can be infinitely many
B
′
s
B's
B
′
s
for which
A
B
=
B
A
AB=BA
A
B
=
B
A
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