Invertible Matrices
NCERT EXERCISE 3.4 • FINAL MCQ
💡 Mastery Tip: Defining Inverses
In matrix algebra, two square matrices $A$ and $B$ are considered inverses of each other if and only if their product, regardless of the order of multiplication, results in the **Identity Matrix ($I$)**.
$AB = BA = I$
If this condition is met, $B$ is the inverse of $A$ (denoted as $A^{-1}$) and $A$ is the inverse of $B$ (denoted as $B^{-1}$).
Multiple Choice
Matrices $A$ and $B$ will be inverse of each other only if:
Evaluating the given options based on the definition of invertible matrices:
- ❌ (A) $AB = BA$: While inverses do commute, this condition alone is not enough. Many non-inverse matrices also commute.
- ❌ (B) $AB = BA = 0$: This describes zero-divisors, which are the opposite of invertible matrices.
- ❌ (C) $AB = 0, BA = I$: Both products must result in the Identity matrix for the inverse to be valid.
- ✅ (D) $AB = BA = I$: This is the exact formal definition of inverse matrices.
Final Answer: (D) AB = BA = I