Let \(M\) and \(N\) be two matrices over \(R\) of order 2. Then, \(M N=N M\) if .......
Let \(M\) and \(N\) be two matrices over \(R\) of order 2. Then, \(M N=N M\) if .......
One of \(M\) and \(N\) is a diagonal matrix
Both \(M\) and \(N\) are diagonal matrices
Both \(M\) and \(N\) are invertible matrices
None of these options are true in general
Solution
Let \(M=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right), N=\left(\begin{array}{ll}p & q \\ r & s\end{array}\right)\)
Then, \(\quad M N=\left(\begin{array}{ll}a p+b r & a q+b s \\ p c+d r & c q+d s\end{array}\right)\)
and \(\quad N M=\left(\begin{array}{cc}a p+q c & p b+q d \\ a r+c s & b r+d s\end{array}\right)\)
Clearly, \(M N=N M\)
When,
\(\begin{aligned}
& b r=0, q c=0 \\
& p c=0, d r=0 \ldots \\
& \text {or, If } \quad b=c=q=r=0 \\
\end{aligned}\)
i.e., both \(M\) and \(N\) must be diagonal matrices.