Let A = ( \beginmatrix 1 & 2 3 & 4 \endmatrix ) and B = ( \beginmatrix a & 0 0 & b \endmatrix ),a,b \in N. Then

there exist infinitely many B' s such that AB=BA

there cannot exist any B such that AB=BA

there exist more then one but finite number of B' s such that AB=BA

there exists exactly one B such that AB=BA

Matrices and Determinants PYQ — JEE Main 2006 · Shift 0 · Q60 with Solution

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Correct answer: D

We are given $A=\begin{pmatrix}1&2\\3&4\end{pmatrix},\qquad B=\begin{pmatrix}a&0\\0&b\end{pmatrix},\quad a,b\in \mathbb N.$ We need to find when $AB=BA.$
Compute $AB$ :

=\begin{pmatrix}a&2b\\3a&4b\end{pmatrix}.$$ 3. Compute $BA$: $$BA=\begin{pmatrix}a&0\\0&b\end{pmatrix}\begin{pmatrix}1&2\\3&4\end{pmatrix} =\begin{pmatrix}a&2a\\3b&4b\end{pmatrix}.$$ 4. Equate corresponding entries of $AB$ and $BA$: $$\begin{pmatrix}a&2b\\3a&4b\end{pmatrix}= \begin{pmatrix}a&2a\\3b&4b\end{pmatrix}.$$ Thus, - from the $(1,2)$ entry: $$2b=2a \implies a=b,$$ - from the $(2,1)$ entry: $$3a=3b \implies a=b.$$ So the necessary and sufficient condition is $$a=b.$$ 5. Therefore, $$B=\begin{pmatrix}a&0\\0&a\end{pmatrix}=aI,$$ where $a\in \mathbb N$. Since $a$ can be any natural number, there are infinitely many such matrices $B$. 6. Check options: - A: false - B: false - C: false - D: true Hence the correct option is $$\boxed{D}.$$

View written solutionFree

Correct answer: D

We are given $A=\begin{pmatrix}1&2\\3&4\end{pmatrix},\qquad B=\begin{pmatrix}a&0\\0&b\end{pmatrix},\quad a,b\in \mathbb N.$ We need to find when $AB=BA.$
Compute $AB$ :

=\begin{pmatrix}a&2b\\3a&4b\end{pmatrix}.$$ 3. Compute $BA$: $$BA=\begin{pmatrix}a&0\\0&b\end{pmatrix}\begin{pmatrix}1&2\\3&4\end{pmatrix} =\begin{pmatrix}a&2a\\3b&4b\end{pmatrix}.$$ 4. Equate corresponding entries of $AB$ and $BA$: $$\begin{pmatrix}a&2b\\3a&4b\end{pmatrix}= \begin{pmatrix}a&2a\\3b&4b\end{pmatrix}.$$ Thus, - from the $(1,2)$ entry: $$2b=2a \implies a=b,$$ - from the $(2,1)$ entry: $$3a=3b \implies a=b.$$ So the necessary and sufficient condition is $$a=b.$$ 5. Therefore, $$B=\begin{pmatrix}a&0\\0&a\end{pmatrix}=aI,$$ where $a\in \mathbb N$. Since $a$ can be any natural number, there are infinitely many such matrices $B$. 6. Check options: - A: false - B: false - C: false - D: true Hence the correct option is $$\boxed{D}.$$