IIT JEE Main Maths -Unit 3- Consistency and solution of linear equations (2 or 3 variables)- Study Notes-New Syllabus

Applications of Determinants and Matrices — Consistency and Solving System of Linear Equations (Cramer’s Rule & Matrix Method)

Introduction

Determinants and matrices are powerful tools for solving systems of linear equations of the form:

$ a_1x + b_1y + c_1z = d_1\\ a_2x + b_2y + c_2z = d_2\\ a_3x + b_3y + c_3z = d_3 $

This system can be represented in matrix form as:

$ A X = B $

where

• \( A = \begin{bmatrix}a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3\end{bmatrix} \) — coefficient matrix
• \( X = \begin{bmatrix}x\\y\\z\end{bmatrix} \) — variable matrix
• \( B = \begin{bmatrix}d_1\\d_2\\d_3\end{bmatrix} \) — constant matrix

Consistency of a System

Let \( A X = B \) represent a system of equations.

• If \( |A| \ne 0 \), the system is consistent and has a unique solution.
• If \( |A| = 0 \), the system may be consistent (infinite solutions) or inconsistent (no solution) depending on \( \text{adj}(A)B = 0 \) or not.

Condition Summary:

 Cramer’s Rule (for 2 or 3 Variables)

If \( |A| \ne 0 \), the system has a unique solution given by:

$ x = \dfrac{|A_x|}{|A|}, \quad y = \dfrac{|A_y|}{|A|}, \quad z = \dfrac{|A_z|}{|A|} $

where \( A_x, A_y, A_z \) are obtained by replacing the 1st, 2nd, and 3rd columns of \( A \) with \( B \), respectively.

Matrix Method

For \( A X = B \):

If \( |A| \ne 0 \), then multiplying both sides by \( A^{-1} \):

\( A^{-1} A X = A^{-1} B \Rightarrow X = A^{-1} B \)

This gives a compact and powerful method to find \( x, y, z \).

Key Takeaways for JEE:

• Use Cramer’s Rule for small systems (2×2 or 3×3).
• Matrix method \( X = A^{-1}B \) is compact and direct for unique solutions.
• Always check \( |A| \) first for consistency.
• If \( |A| = 0 \) and \( \text{adj}(A)B \ne 0 \), → No solution (inconsistent).
• If \( |A| = 0 \) and \( \text{adj}(A)B = 0 \), → Infinite solutions (consistent).