Matrices and Determinants Class 12: Properties and Solved Examples

Quick answer: Matrices and Determinants together carry 8–10 marks in Class 12 boards, and the 5-mark ‘solve by matrix method’ question is almost guaranteed. Here is the complete toolkit.

Matrices

• Order m × n; equal only if same order and elements.
• Types: row, column, square, diagonal, scalar, identity I, zero, upper/lower triangular.
• Addition element-wise; multiplication AB needs columns of A = rows of B; AB ≠ BA in general.
• Transpose: (AB)ᵀ = BᵀAᵀ. Symmetric: Aᵀ = A; skew-symmetric: Aᵀ = −A (diagonal zero).
• Any square matrix = ½(A + Aᵀ) + ½(A − Aᵀ).

Determinants

• 2×2: ad − bc. 3×3: expand along any row/column with sign pattern + − +.
• Properties: swapping two rows changes sign; two identical rows → 0; a common factor of a row can be taken out; row operations R₁ → R₁ + kR₂ leave it unchanged.
• Area of triangle = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; collinear if 0.

Adjoint and inverse

• adj A = transpose of the cofactor matrix; A(adj A) = |A| I
• A⁻¹ = adj A / |A|, exists only if |A| ≠ 0 (non-singular)
• (AB)⁻¹ = B⁻¹A⁻¹; |adj A| = |A|ⁿ⁻¹

Matrix method (AX = B)

1. Write A (coefficients), X (variables), B (constants).
2. Find |A|; if ≠ 0, unique solution X = A⁻¹B.
3. Compute adj A, then A⁻¹, then multiply.

Example

2x + 3y = 5, x − y = 0. A = [[2, 3],[1, −1]], |A| = −5, adj A = [[−1, −3],[−1, 2]], A⁻¹ = −1/5 × adj A. X = A⁻¹B = [[1],[1]] → x = 1, y = 1.

Common mistakes

• Writing |A| as a matrix instead of a number
• Forgetting to transpose the cofactor matrix for adjoint
• Multiplying matrices in the wrong order

Related: Differentiation formulas.

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Frequently asked questions

Are these Matrices and Determinants Class 12 notes enough for the exam?

They cover every formula, definition and question type that appears in school and board papers for this chapter. Use them with the NCERT examples, exercise questions and the last five years' papers to be fully prepared.

How should I revise Matrices and Determinants Class 12?

Revise the notes after 1 day, 7 days and 30 days. Solve five questions each time instead of only reading, and keep an error list of the steps you get wrong.

What if I get stuck on a matrices class 12 question?

Type or photograph the exact question on DoubtPay AI, choose your class and language, and get a step-by-step answer with a diagram. You can also ask for a hint or have your own attempt checked. Packs start at ₹9 for 15 answers.