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6.Solving Linear Equations
Class 12 • Mathematics • NCERT
Solving Linear Equations Using Matrices
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In this lesson, we solve systems of linear equations using matrices by applying the inverse of the coefficient matrix, exactly as prescribed in the NCERT Class 12 syllabus.
Lesson Objectives
- Form a system of equations in matrix form.
- Apply inverse matrix method to solve equations.
- Understand the condition for existence of solution.
- Solve NCERT-based exam questions confidently.
1. System of Linear Equations
A system of linear equations consists of two or more linear equations involving the same variables.
Example:
x + y = 4
2x − y = 1
x + y = 4
2x − y = 1
2. Matrix Representation
Any system of linear equations can be written in matrix form as:
AX = B
where A is the coefficient matrix, X is the column matrix of variables, and B is the constant matrix.
3. Condition for Solvability
The system of equations has a unique solution if the determinant of matrix A is not zero.
|A| ≠ 0
4. Inverse Matrix Method
If |A| ≠ 0, then A⁻¹ exists and the solution of the system AX = B is given by:
X = A⁻¹B
5. NCERT Solved Example
Solve the system:
x + 2y = 5
3x + 4y = 11
x + 2y = 5
3x + 4y = 11
Matrix form:
A = |1 2|
|3 4|
X = |x|
|y|
B = |5|
|11|
|A| = (1×4 − 2×3) = −2 ≠ 0
⇒ Unique solution exists
6. Important NCERT Points
• Applicable only when determinant is non-zero
• Coefficient matrix must be square
• Method gives only unique solutions
• Based on adjoint and inverse of matrices
• Coefficient matrix must be square
• Method gives only unique solutions
• Based on adjoint and inverse of matrices
Practice Questions (NCERT Based)
- Write the matrix form of a system of linear equations.
- State the condition for existence of inverse of a matrix.
- What is the formula to solve AX = B?
- Is inverse method applicable when |A| = 0?
- What type of matrix must A be?
- Name the matrix containing constants.
- Which chapter introduces this method?
- Does this method give infinite solutions?
- Is this method part of NCERT Class 12 syllabus?
- Which concept is prerequisite for this method?
✅ Show Answer Key
- AX = B
- |A| ≠ 0
- X = A⁻¹B
- No
- Square matrix
- Constant matrix B
- Matrices
- No
- Yes
- Adjoint and inverse of matrix
© Aviate Learning – Solving Linear Equations Using Matrices (NCERT Class 12)