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5.Adjoint & Inverse of Matrix
Class 12 • Mathematics • NCERT
Adjoint & Inverse of a Matrix
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This lesson explains the adjoint and inverse of a matrix as prescribed by NCERT Class 12. These concepts are essential for solving matrix equations and systems of linear equations.
Lesson Objectives
- Understand the meaning of adjoint of a matrix.
- Learn how to find inverse using adjoint.
- Know the condition for existence of inverse.
- Solve NCERT exam-type problems.
1. Adjoint of a Matrix
The adjoint of a square matrix A, denoted by adj(A), is the transpose of the matrix formed by cofactors of A.
adj(A) = (Cofactor matrix of A)T
2. Steps to Find Adjoint
Step 1: Find all minors of A
Step 2: Convert minors into cofactors
Step 3: Arrange cofactors in matrix form
Step 4: Take transpose
Step 2: Convert minors into cofactors
Step 3: Arrange cofactors in matrix form
Step 4: Take transpose
3. Inverse of a Matrix
The inverse of a matrix A exists if and only if |A| ≠ 0. It is denoted by A⁻¹.
A⁻¹ = (1 / |A|) adj(A)
4. Important NCERT Properties
A · adj(A) = |A| I
adj(A) · A = |A| I
(A⁻¹)⁻¹ = A
adj(A) · A = |A| I
(A⁻¹)⁻¹ = A
5. Simple NCERT Example
For A = |1 2|
|3 4|
|3 4|
|A| = −2 ≠ 0
adj(A) = |4 −2|
|−3 1|
A⁻¹ = (1/−2) adj(A)
Practice Questions (NCERT)
- Define adjoint of a matrix.
- When does the inverse of a matrix exist?
- Write the formula for A⁻¹.
- What is adj(A) if A is identity matrix?
- Is inverse defined for non-square matrices?
- State A · adj(A).
- If |A| = 0, does A⁻¹ exist?
- What is the inverse of identity matrix?
- What is the order of adj(A)?
- Is this topic part of NCERT Class 12 syllabus?
✅ Show Answer Key
- Transpose of cofactor matrix
- When determinant ≠ 0
- A⁻¹ = (1/|A|) adj(A)
- Identity matrix
- No
- |A| I
- No
- Identity matrix
- Same as A
- Yes
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