Q.Matrices $A$ and $B$ will be inverse of each other only if (A) $AB = BA$ (B) $AB = BA = 0$ (C) $AB = 0,\ BA = I$ (D) $AB = BA = I$

The key idea is the definition of inverse matrices: two square matrices are inverses if their product (in either order) equals the identity matrix.

Step 1: For matrices $A$ and $B$ to be inverses, they must be square matrices of the same order.

Step 2: By definition, $B$ is the inverse of $A$ if and only if $AB = I$ and $BA = I$, where $I$ is the identity matrix. …

The idea of an inverse matrix comes directly from the concept of a reciprocal in numbers. For a real number $a$, its inverse $a^{-1}$ satisfies $a \cdot a^{-1} = a^{-1} \cdot a = 1$. For matrices, the "1" is replaced by the identity matrix $I$, and multiplication must work both ways because matrix multiplication is not commutative in general.

A square matrix $A$ is said to be invertible (or non-singular) if there exists another square matrix $B$ of the same order such that:

When this holds, we write $B = A^{-1}$.

Now let's examine each option carefully.

1. Option (A): $AB = BA$

   This only says that $A$ and $B$ commute. Commuting is a property many matrix pairs have — for example, any two diagonal matrices commute — but that does not make them inverses. If $A$ and $B$ commute but their product is not $I$, they are not inverses. So this condition is necessary but far from sufficient.

2. Option (B): $AB = BA = 0$

   If the product of two matrices is the zero matrix, then neither matrix can be invertible (unless one of them is zero, which is never invertible). An invertible matrix multiplied by its inverse gives $I$, not $0$. This condition describes a pair of matrices that are "zero divisors", not inverses.

3. Option (C): $AB = 0,\ BA = I$

   This is contradictory. If $AB = 0$, then multiplying on the left by $A^{-1}$ (if it existed) would give $B = 0$, but then $BA = 0$, not $I$. More directly, if $BA = I$, then $B$ is a left-inverse of $A$. But for square matrices, a left-inverse is automatically a right-inverse — meaning $AB$ must also equal $I$. So $AB = 0$ cannot happen if $BA = I$. This option is impossible.

4. Option (D): $AB = BA = I$ …

Method: Applying the definition of the inverse of a matrix

Use this reasoning whenever a question asks which condition makes two matrices inverses of each other.

Steps

Step 1: Recall the defining condition.

$B$ is the inverse of $A$ (both square, same order) exactly when

The identity $I$ plays the role the number $1$ plays for reciprocals.

Step 2: Test each candidate condition against the definition. …

Common Mistakes

Mistake 1: Choosing $AB = BA$ (mere commuting).

Why it's wrong: commuting matrices need not be inverses — the product must equal the identity, not just agree in both orders. Correct approach: require $AB = BA = I$.

Mistake 2: Thinking $AB = 0,\ BA = I$ is possible.

Why it's wrong: if $BA = I$ then $A$ is invertible, and then $AB = 0$ would force $B = 0$, contradicting $BA = I$. Correct approach: recognise this option as impossible. …