Why does $A B = O$ not imply $A = O$ or $B = O$ in Matrix Algebra?

Matrices represent spatial transformations, not simple numeric magnitudes. A non-zero matrix $B$ can transform vectors into the null space of another non-zero matrix $A$, resulting in a zero product matrix $O$. Hence, never divide by a matrix or cancel terms without confirming non-singularity ($|A| \neq 0$).

How do I quickly find higher powers $A^n$ in JEE Advanced questions?

First, evaluate $A^2$ and $A^3$ to look for a predictable arithmetic sequence in entries. If $A$ is diagonal or triangular, pattern recognition or mathematical induction works best. Alternatively, if $A$ satisfies $A^2 - \text{tr}(A)A + |A|I = O$, use the Cayley-Hamilton theorem to express $A^n$ as a linear combination of $A$ and $I$.

What is the difference between a skew-symmetric matrix and an orthogonal matrix?

A skew-symmetric matrix satisfies $A^T = -A$ (with all principal diagonal elements equal to $0$), whereas an orthogonal matrix satisfies $A^T A = I$, meaning its inverse equals its transpose ($A^{-1} = A^T$).

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Question #1JEE Main 2024

If $A$ is a $3 \times 3$ real matrix such that $|A| = 4$ , then the value of $|2 \text{adj}(A)|$ is equal to:

Solution Locked

Step-by-Step Solution & Concept

Step 1

Apply the determinant scalar property $|k B| = k^n |B|$ for matrix order $n = 3$ , giving $|2 \text{adj}(A)| = 2^3 |\text{adj}(A)| = 8 |\text{adj}(A)|$ .

Step 2

Use the adjugate determinant identity $|\text{adj}(A)| = |A|^{n-1} = |A|^{3-1} = |A|^2$ .

Step 3

Substitute $|A| = 4$ into the expression: $|2 \text{adj}(A)| = 8 \times (4)^2 = 8 \times 16 = 128$ .

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Question #2JEE Main 2023

Let $A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$ . Then $A^{50}$ is equal to:

Solution Locked

Step-by-Step Solution & Concept

Step 1

Compute $A^2$ to discover the matrix power pattern: $A^2 = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$ .

Step 2

Compute $A^3 = A^2 A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix}$ , establishing by mathematical induction that $A^n = \begin{bmatrix} 1 & 0 \\ n & 1 \end{bmatrix}$ .

Step 3

Substituting $n = 50$ directly yields $A^{50} = \begin{bmatrix} 1 & 0 \\ 50 & 1 \end{bmatrix}$ .

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Mastering Matrices: Formulas, Traps & Exam Strategy | ExamiFly

Chapter Overview

Core Subtopics & Syllabus Breakdown

Key Formulas & High-Yield Equations

Inverse of a Square Matrix

Adjugate Fundamental Property

Determinant of Adjugate Matrix

Symmetric and Skew-Symmetric Decomposition

Orthogonal Matrix Identity

Cayley-Hamilton Characteristic Equation

Common Mistakes & Traps to Avoid

Past Year Question (PYQ) Trends

Solved High-Yield Sample Questions

Frequently Asked Questions

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