Matrix Inverse

• The definition of inverse matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I.
• To compute A⁻¹ for 2×2 and 3×3 matrices using the adjugate method: A⁻¹ = (1/|A|) adj(A).
• When an inverse exists (|A| ≠ 0) and when it does not (singular matrix).
• To solve linear systems via X = A⁻¹B and compare with row-reduction methods.
• To understand inverse of products: (AB)⁻¹ = B⁻¹A⁻¹ (reverse order).

Level 1 — Foundations

Verbal: For a square matrix A, the inverse A⁻¹ is the unique matrix (if it exists) that undoes multiplication by A — like reciprocal for numbers.

Existence: A⁻¹ exists ⟺ |A| ≠ 0 ⟺ A is non-singular.

2×2 formula: A = | a b |, A⁻¹ = (1/(ad−bc)) | d −b | | c d | | −c a |

Identity check: AA⁻¹ = I and A⁻¹A = I where I has 1s on diagonal.

Adjugate: adj(A) is transpose of cofactor matrix. For invertible A: A⁻¹ = adj(A)/|A|.

Level 2 — JEE / NEET depth

Steps for 3×3 inverse (NCERT):

1. Compute |A|. If zero, stop — no inverse.
2. Find cofactor Cᵢⱼ of each entry.
3. Form cofactor matrix, transpose → adj(A).
4. A⁻¹ = (1/|A|) adj(A).

Properties for JEE:

Solving equations: AX = B → X = A⁻¹B (if A invertible). For n equations n unknowns, this is equivalent to Cramer's rule when applicable.

Elementary matrices: Row operations correspond to multiplying by elementary matrices; inverse operations reverse the steps.

Find inverse of a 2×2 matrix

A = | 4  3 |
    | 2  1 |

Step 1 — |A| = (4)(1) − (3)(2) = 4 − 6 = −2 ≠ 0 → inverse exists.
Step 2 — Swap diagonal, negate off-diagonal:
         adj(A) = |  1  −3 |
                  | −2   4 |
Step 3 — A⁻¹ = (1/(−2)) |  1  −3 | = | −1/2   3/2 |
                        | −2   4 |   |  1    −2  |
Step 4 — Check: AA⁻¹ = | 4  3 | | −1/2  3/2 | = | 1  0 | ✓
                       | 2  1 | |  1   −2  |   | 0  1 |

Solve 2×2 system using inverse

System: 2x + y = 5,  4x + 3y = 11.

Step 1 — Matrix form: | 2  1 | | x | = |  5 |
                       | 4  3 | | y |   | 11 |
         A = above 2×2, X = column (x,y)ᵀ, B = (5,11)ᵀ.
Step 2 — From previous example structure: |A| = 2, A⁻¹ = (1/2)| 3 −1 |
                                                              | −4  2 |
Step 3 — X = A⁻¹B = (1/2)| 3 −1 | | 5 | = (1/2)| 4 | = | 2 |
                          | −4  2 | | 11|       | 6 |   | 3 |
Step 4 — Solution: x = 2, y = 3. Verify in both equations.

• Does a 2×3 matrix have an inverse? Explain.
• Find A⁻¹ if A = | 1 2 |.
• If A and B are invertible, simplify (A⁻¹B)⁻¹.
• For what k is | k 1 | singular?
• Stretch: Show (Aᵀ)⁻¹ = (A⁻¹)ᵀ for invertible A.

NCERT Chapter 3 link: Inverse matrices connect directly to solving systems AX = B when A is square and non-singular. NCERT shows adjoint method for 3×3; for larger orders, row reduction is practical though beyond standard Class 12 requirement.

Exam connections: Typical JEE problems give parametric matrix and ask for k such that A⁻¹ exists (|A| ≠ 0). Board questions may ask to solve 3×3 system — show adjoint steps clearly. Prove (AB)⁻¹ = B⁻¹A⁻¹ once using definition AA⁻¹ = I — reusable in proof-based questions.

Study strategy: Always compute |A| first — saves time when singular. For 2×2, memorise swap-and-divide pattern but derive on rough sheet if unsure. Connect to coordinate geometry: inverse of transformation matrix reverses the mapping.

When studying Matrix Inverse within Matrices & Determinants, start by listing every formula and definition on one page without looking at the textbook. Compare your list to NCERT — missing items indicate gaps to fix immediately. Work through at least two NCERT Examples for this section with steps written in full; examiners award method marks even when arithmetic slips.

For board exams (CBSE), long answers benefit from a clear structure: definition → explanation → diagram or formula → example → brief conclusion. Underline key terms. For JEE Main and NEET, prioritise conceptual traps and quick calculation paths; timed mixed quizzes of 10 questions after revision simulate exam pressure.

Cross-topic link: Coordinate geometry and vectors often combine with matrices; calculus links to physics kinematics problems.

Spaced revision: Review this note at 1 day, 3 days, and 7 days after first study. Attempt the Quick check questions closed-book, then open the Practice tab for graded reinforcement. Maintain an error log — repeated mistake patterns reveal whether the issue is concept, formula recall, or careless reading.

Diagram and terminology drill: For Mathematics, redraw key figures from memory and define every labelled part in one sentence. Vocabulary precision prevents mark loss in descriptive answers — use NCERT terms exactly as printed in the textbook.

Revision tip: Link this topic to adjacent Class 12 chapters before attempting mixed practice.

Open the Practice tab for graded questions on Matrix Inverse.