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Q1:
Matrices A and B will be inverse of each other only if
Solution :
Theoretical Foundation of Matrix Inversion
Two square matrices $A$ and $B$ of the same order $n \times n$ are defined as inverses of each other if and only if their product results in the Identity matrix $I$ of the same order. This relationship is derived from the definition of the multiplicative inverse in linear algebra.
Step 1: Establishing the Mathematical Condition
By the definition of an invertible matrix [Per the definition of Matrix Inversion in Linear Algebra], a matrix $B$ is the inverse of matrix $A$ if the following criteria are met simultaneously:
- $AB = I$
- $BA = I$
Where $I$ denotes the Identity matrix of order $n$.
Step 2: Analytical Representation
If $B$ is the inverse of $A$, it is denoted as $B = A^{-1}$. Substituting this into the conditions established in Step 1:
$A \cdot A^{-1} = I$
$A^{-1} \cdot A = I$
Step 3: Geometric Visualization of Identity Matrix Mapping
The following SVG represents the identity transformation $I$ for a $2 \times 2$ matrix, where the basis vectors $\hat{i}$ and $\hat{j}$ remain fixed in the Cartesian plane.
Step 4: Summary Table of Properties
| Condition | Mathematical Expression | Logical Justification |
|---|---|---|
| Commutativity with Inverse | $AB = BA = I$ | [Definition of Multiplicative Inverse] |
| Determinant Requirement | $|A| \neq 0$ | [Requirement for Non-singularity] |
Final Solution: Matrices $A$ and $B$ are inverses of each other if and only if $AB = BA = I$, where $I$ is the identity matrix of the same order.
CBSE Solutions for Class 12 Mathematics Matrices
Chapters in CBSE - Class 12 Mathematics
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