When should I use Cramer's Rule versus Matrix Inversion for solving systems of equations in JEE Main?

In timed competitive exams like JEE Main, Cramer's Rule using determinants $(\Delta, \Delta_x, \Delta_y, \Delta_z)$ is significantly faster and less prone to calculation errors than calculating $A^{-1} = \frac{\text{adj}(A)}{\det(A)}$. Always use Cramer's Rule when analyzing parameters like $\lambda$ and $\mu$.

Does performing simultaneous row transformations alter the determinant value?

Yes, if executed incorrectly. Operations of the form $R_i \to R_i + k R_j$ leave $\det(A)$ unchanged. However, never alter the leading target row simultaneously in cyclic operations like $R_1 \to R_1 - R_2$ and $R_2 \to R_2 - R_1$ in the exact same step, as this violates linear independence.

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Question #1JEE Main 2024

If $A$ is a $3 \times 3$ matrix such that $\det(A) = 4$ , then what is the value of $\det(3 \cdot \text{adj}(2A))$ ?

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Step-by-Step Solution & Concept

Step 1

Apply scalar rule $\det(k B) = k^n \det(B)$ with $n = 3$ and $k = 3$ , giving $\det(3 \cdot \text{adj}(2A)) = 3^3 \det(\text{adj}(2A))$ .

Step 2

Use the adjoint property $\det(\text{adj}(M)) = (\det(M))^{n-1}$ for $n=3$ , yielding $\det(\text{adj}(2A)) = (\det(2A))^2$ .

Step 3

Evaluate $\det(2A) = 2^3 \det(A) = 8 \times 4 = 32 = 2^5$ . Substituting back gives $3^3 \cdot (2^5)^2 = 3^3 \cdot 2^{10}$ .

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Question #2JEE Main 2023

Consider the system of linear equations: $x + y + z = 6$ , $x + 2y + 3z = 10$ , $x + 2y + \lambda z = \mu$ . The system has infinitely many solutions if:

Solution Locked

Step-by-Step Solution & Concept

Step 1

Compute the main determinant $D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \lambda \end{vmatrix}$ . Applying $R_3 \to R_3 - R_2$ yields $D = \lambda - 3$ .

Step 2

For non-unique solutions, set $D = 0 \implies \lambda - 3 = 0 \implies \lambda = 3$ .

Step 3

Substitute $\lambda = 3$ into the third equation to get $x + 2y + 3z = \mu$ . Comparing with the second equation $x + 2y + 3z = 10$ , consistency requires $\mu = 10$ .

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Mastering Determinants: Formulas, Traps & Exam Strategy | ExamiFly

Chapter Overview

Core Subtopics & Syllabus Breakdown

Key Formulas & High-Yield Equations

Multiplicative Determinant Property

Scalar Multiplication Rule

Determinant of Inverse Matrix

Determinant of Adjoint Matrix

Skew-Symmetric Matrix Determinant

Common Mistakes & Traps to Avoid

Past Year Question (PYQ) Trends

Solved High-Yield Sample Questions

Frequently Asked Questions

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