Maharashtra State Board · Class 12 · Mathematics · Part I Chapter 2

Matrices and Determinants — Formula Sheet

Board Formulas

12 formulas

1.Matrix Multiplication★
$(A B)_{ij} = \sum_{k} a_{ik} b_{k j}$
Row × column dot product. Defined only when columns of A = rows of B.
2.Non-commutativity
$A B \neq = B A (in general)$
Matrix multiplication is NOT commutative except in special cases.
3.Transpose
$(A^{T})_{ij} = A_{j i}, (A B)^{T} = B^{T} A^{T}$
Reversal rule for transpose of product.
4.Determinant of 3×3★
$∣ A ∣ = a_{11} (a_{22} a_{33} - a_{23} a_{32}) - a_{12} (a_{21} a_{33} - a_{23} a_{31}) + a_{13} (a_{21} a_{32} - a_{22} a_{31})$
Cofactor expansion along first row. Use any row/column with zeros for simplicity.
5.Properties of Determinants★
$∣ A^{T} ∣ = ∣ A ∣, ∣ A B ∣ = ∣ A ∣∣ B ∣, ∣ k A ∣ = k^{n} ∣ A ∣ (n -order matrix)$
Row swap changes sign; adding multiple of one row to another leaves |A| unchanged.
6.Adjoint (adjugate)★
$adj (A) = C^{T}, C_{ij} = cofactor of a_{ij}$
Transpose of the cofactor matrix. Basis of inverse formula.
7.A · adj(A) = |A|·I★
$A \cdot adj (A) = ∣ A ∣ \cdot I_{n}$
Fundamental identity linking adjoint and determinant. Use to verify computed adjoint.
8.Inverse of a Matrix★
$A^{- 1} = \frac{1}{∣ A ∣} adj (A), if ∣ A ∣ \neq = 0$
A is invertible iff |A| ≠ 0. Then AA⁻¹ = A⁻¹A = I.
9.Solving Linear System — Matrix Method★
$A X = B \Rightarrow X = A^{- 1} B (if ∣ A ∣ \neq = 0)$
Rewrite the system as AX = B, invert A, multiply by B.
10.Cramer's Rule★
$x_{j} = \frac{Δ _{j}}{Δ}, Δ = ∣ A ∣, Δ_{j} = ∣ A with column j replaced by B ∣$
Alternative to A⁻¹B method. Compact for 2×2 and 3×3 systems.
11.Consistency Criterion
$∣ A ∣ \neq = 0 : unique; ∣ A ∣ = 0, (adj A) B = 0 : infinite; ∣ A ∣ = 0, (adj A) B \neq = 0 : no solution$
State all three cases in theory questions on consistency.
12.Elementary Row Operations
$R_{i} \to R_{i} + k R_{j}, R_{i} \leftrightarrow R_{j}, R_{i} \to k R_{i}$
Preserve solution set. Used in row reduction / Gauss elimination.