Exercise 1
Matrices A and B will be inverse of each other only if (A) (C)
Not cross-checked
This solution has not been cross-checked against the answer printed in NCERT.
NCERT’s answer
D
Use the definition of the inverse of a matrix: if \(\displaystyle \mathrm{A}\) and \(\displaystyle \mathrm{B}\) are square matrices of the same order \(\displaystyle n\), then \(\displaystyle \mathrm{B}\) is called the inverse of \(\displaystyle \mathrm{A}\) (and \(\displaystyle \mathrm{A}\) the inverse of \(\displaystyle \mathrm{B}\)) if
\[\mathrm{AB}=\mathrm{BA}=\mathrm{I},\]
where \(\displaystyle \mathrm{I}\) is the identity matrix of order \(\displaystyle n\). Note that the definition demands both products, and demands that each of them be \(\displaystyle \mathrm{I}\) — that is the part usually dropped.
Check each option against this.
(A)
\(\displaystyle \mathrm{AB}=\mathrm{BA}\) says only that \(\displaystyle \mathrm{A}\) and \(\displaystyle \mathrm{B}\) commute; it does not say what the common product is. For instance \(\displaystyle \mathrm{A}=\mathrm{B}=\left[\begin{array}{rr}2 & 0\\ 0 & 2\end{array}\right]\) gives \(\displaystyle \mathrm{AB}=\mathrm{BA}=\left[\begin{array}{rr}4 & 0\\ 0 & 4\end{array}\right]\neq\mathrm{I}\), so these commuting matrices are not inverses of each other. Not sufficient.
(B)
\(\displaystyle \mathrm{AB}=\mathrm{BA}=0\) is the opposite situation: taking \(\displaystyle \mathrm{A}=\mathrm{B}=\mathrm{O}\) satisfies it, and the zero matrix has no inverse. Indeed a zero product can never be \(\displaystyle \mathrm{I}\), since \(\displaystyle \mathrm{O}\neq\mathrm{I}\).
(C)
\(\displaystyle \mathrm{AB}=0,\ \mathrm{BA}=\mathrm{I}\) cannot happen at all for square matrices. If \(\displaystyle \mathrm{BA}=\mathrm{I}\), then multiplying \(\displaystyle \mathrm{AB}=\mathrm{O}\) on the left by \(\displaystyle \mathrm{B}\) gives \(\displaystyle (\mathrm{BA})\mathrm{B}=\mathrm{B}\,\mathrm{O}\), i.e. \(\displaystyle \mathrm{IB}=\mathrm{O}\), so \(\displaystyle \mathrm{B}=\mathrm{O}\); but then \(\displaystyle \mathrm{BA}=\mathrm{O}\neq\mathrm{I}\), a contradiction. The two conditions are inconsistent.
(D)
\(\displaystyle \mathrm{AB}=\mathrm{BA}=\mathrm{I}\) is exactly the definition, so it is both necessary and sufficient.
The correct option is \(\displaystyle (\mathrm{D})\ \mathrm{AB}=\mathrm{BA}=\mathrm{I}\).