Is it correct to say: Assuming AB = BA and cancelling matrices?

Matrix multiplication is NOT commutative in general. And you cannot cancel unless matrix is invertible.

Is it correct to say: Writing determinant of 3×3 as ad − bc?

That's the 2×2 formula. Use full cofactor expansion for 3×3.

Is it correct to say: Confusing minor and cofactor?

Minor M_{ij} = det of submatrix. Cofactor C_{ij} = (−1)^{i+j}·M_{ij}. Include the sign!

Is it correct to say: Writing adj(A) = C directly (not transposed)?

adj(A) = C^T. Transpose the cofactor matrix.

Is it correct to say: Applying |kA| = k|A| (missing the exponent n)?

|kA| = k^n |A| where n is order of the matrix. For 3×3, |3A| = 27|A|.

Is it correct to say: Using matrix inversion when |A| = 0?

Then A⁻¹ doesn't exist. Use rank method or check consistency with augmented matrix.

Matrices and Determinants

Matrix algebra, determinants, adjoint, inverse, system of linear equations — Maharashtra HSC Maths Ch 2

📐 12 formulas✏️ 3 examples🎯 5 practice⚖️ 8 marks🏫 Maharashtra State Board📚 Class 12✓ 2025–26 syllabus

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Board Exam Tips

→Solving a system of 3 linear equations by matrix inversion (X = A⁻¹B) is a Maharashtra HSC 4-marker every year.
→Properties of determinants — expand a 3×3 determinant using column/row operations to show it equals a given expression — 3-marker.
→Adjoint of a 3×3 matrix and verification of A·adj(A) = |A|·I — 4-mark HSC classic.
→Cramer's rule (Δ₁/Δ, Δ₂/Δ, Δ₃/Δ) is asked as an alternative method — mention which method you are using.
→State the conditions for consistency of a linear system (unique, infinite, no solution) based on rank / determinant sign.

📐 Formulas(12)

Matrix Multiplication★ Board fav

(A B)_{ij} = \sum_{k} a_{ik} b_{k j}

Non-commutativity

A B \neq = B A (in general)

Transpose

(A^{T})_{ij} = A_{j i}, (A B)^{T} = B^{T} A^{T}

Determinant of 3×3★ Board fav

∣ A ∣ = a_{11} (a_{22} a_{33} - a_{23} a_{32}) - a_{12} (a_{21} a_{33} - a_{23} a_{31}) + a_{13} (a_{21} a_{32} - a_{22} a_{31})

Properties of Determinants★ Board fav

∣ A^{T} ∣ = ∣ A ∣, ∣ A B ∣ = ∣ A ∣∣ B ∣, ∣ k A ∣ = k^{n} ∣ A ∣ (n -order matrix)

Adjoint (adjugate)★ Board fav

adj (A) = C^{T}, C_{ij} = cofactor of a_{ij}

A · adj(A) = |A|·I★ Board fav

A \cdot adj (A) = ∣ A ∣ \cdot I_{n}

Inverse of a Matrix★ Board fav

A^{- 1} = \frac{1}{∣ A ∣} adj (A), if ∣ A ∣ \neq = 0

Solving Linear System — Matrix Method★ Board fav

A X = B \Rightarrow X = A^{- 1} B (if ∣ A ∣ \neq = 0)

Cramer's Rule★ Board fav

x_{j} = \frac{Δ _{j}}{Δ}, Δ = ∣ A ∣, Δ_{j} = ∣ A with column j replaced by B ∣

Consistency Criterion

∣ A ∣ \neq = 0 : unique; ∣ A ∣ = 0, (adj A) B = 0 : infinite; ∣ A ∣ = 0, (adj A) B \neq = 0 : no solution

Elementary Row Operations

R_{i} \to R_{i} + k R_{j}, R_{i} \leftrightarrow R_{j}, R_{i} \to k R_{i}

✏️ Solved Examples

1Solved Exampleeasy3 steps

Evaluate the determinant |A| where A = [[1, 2, 3],[4, 5, 6],[7, 8, 10]].

Expand along the first row.

∣ A ∣ = 1 (5 \cdot 10 - 6 \cdot 8) - 2 (4 \cdot 10 - 6 \cdot 7) + 3 (4 \cdot 8 - 5 \cdot 7)

2Solved Exampleboard5 steps

Solve the system: 2x + 3y − z = 5, x − y + 2z = 3, 3x + y + z = 8 using matrix inversion method.

Write in matrix form AX = B.

A = 213 3 - 1 1 - 1 21, B = 538

3Solved ExampleHOTS3 steps

If A is a 3×3 matrix with |A| = 4, find |adj(A)| and |3A|.

Property: for n×n matrix, |adj(A)| = |A|^(n−1).

∣ adj (A) ∣ = ∣ A ∣^{n - 1} = ∣ A ∣^{2} = 16

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Assuming AB = BA and cancelling matrices
✓Matrix multiplication is NOT commutative in general. And you cannot cancel unless matrix is invertible.
2
Writing determinant of 3×3 as ad − bc
✓That's the 2×2 formula. Use full cofactor expansion for 3×3.
3
Confusing minor and cofactor
✓Minor M_{ij} = det of submatrix. Cofactor C_{ij} = (−1)^{i+j}·M_{ij}. Include the sign!
4
Writing adj(A) = C directly (not transposed)
✓adj(A) = C^T. Transpose the cofactor matrix.
5
Applying |kA| = k|A| (missing the exponent n)
✓|kA| = k^n |A| where n is order of the matrix. For 3×3, |3A| = 27|A|.
6
Using matrix inversion when |A| = 0
✓Then A⁻¹ doesn't exist. Use rank method or check consistency with augmented matrix.

🎯 Practice Yourself

🎯Practice Yourself5 questions

Q1
If A = [[1, 2],[3, 4]], find A⁻¹.
Q2
Solve x + 2y = 4, 3x − y = 5 using Cramer's rule.
Q3
If |A| = 2 for a 3×3 matrix, find |2A⁻¹|.
Q4
Show that determinant of a skew-symmetric matrix of odd order is zero.
Q5
Find k so that the system kx + y = 1, x + ky = 1 has no unique solution.

📝 Notes

Matrices and Determinants — Maharashtra HSC Overview

Chapter 2 of the Balbharati Class 12 Mathematics textbook covers matrix algebra, determinants, adjoint, inverse, and applications to linear systems.

Maharashtra syllabus specifics

Solution of a 3×3 linear system by matrix inversion — asked every year for 4 marks. Balbharati prefers this over Gaussian elimination.
Cramer's rule is a Balbharati-specific alternative — HSC often gives the choice.
Properties of determinants with column/row operations to simplify — a favourite 3-marker.
Consistency of a system — unique / infinite / no solution — using |A| and adj(A)·B — Maharashtra-specific theoretical question.
Adjoint verification A·adj(A) = |A|·I is asked as a 3-marker in Section C.

Strategy

For 3×3 determinants, cofactor expansion along the row/column with the most zeros saves effort.
When solving AX = B:
- Compute |A| first. If ≠ 0, proceed with inversion.
- Compute cofactors and adj(A).
- Verify A·adj(A) = |A|·I.
- Then X = adj(A)·B / |A|.
Cramer's rule is faster for 2×2 and 3×3 but only works when |A| ≠ 0.