CBSE · Class 12 · Mathematics · Chapter 3

Matrices and Determinants — Formula Sheet

Board Formulas

14 formulas

1.Matrix Addition
$(A + B)_{ij} = A_{ij} + B_{ij}$
Only defined when A and B have the SAME order. Add corresponding entries.
2.Scalar Multiplication
$(k A)_{ij} = k \cdot A_{ij}$
Multiply every entry by the scalar. Does not affect the order of the matrix.
3.Matrix Multiplication★
$(A B)_{ij} = \sum_{k = 1}^{n} A_{ik} B_{k j}$
Defined only when (columns of A) = (rows of B). Result has (rows of A) × (columns of B). NOT commutative.
4.Transpose★
$(A^{T})_{ij} = A_{j i}, (A B)^{T} = B^{T} A^{T}$
Rows become columns. Note the REVERSAL of order in (AB)^T = B^T A^T.
5.Determinant of 2×2★
$det (a c b d) = a d - b c$
Product of main diagonal minus product of anti-diagonal.
6.Determinant of 3×3 (Cofactor Expansion)★
$∣ A ∣ = a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}, C_{ij} = (- 1)^{i + j} M_{ij}$
Expand along ANY row or column — pick the one with most zeros. M_{ij} = minor after deleting row i, column j.
7.Product Rule for Determinants
$∣ A B ∣ = ∣ A ∣ \cdot ∣ B ∣$
|A| behaves multiplicatively — very useful for showing det ≠ 0 without expanding AB.
8.Adjoint★
$adj (A) = (C_{ij})^{T}$
Transpose of the cofactor matrix. Row i of C becomes column i of adj(A).
9.Fundamental Identity
$A \cdot adj (A) = adj (A) \cdot A = ∣ A ∣ I$
Sanity check on adj(A). If this product is not |A|·I, you made a sign or minor error.
10.Inverse via Adjoint★
$A^{- 1} = \frac{1}{∣ A ∣} adj (A), ∣ A ∣ \neq = 0$
A is invertible ⇔ |A| ≠ 0. If |A| = 0, A is called SINGULAR (no inverse).
11.Matrix Method for Linear System★
$A X = B \Rightarrow X = A^{- 1} B, ∣ A ∣ \neq = 0$
Only valid when |A| ≠ 0. If |A| = 0 and (adj A)·B ≠ 0, system is inconsistent.
12.Consistency Criteria
$∣ A ∣ \neq = 0 \Rightarrow unique; ∣ A ∣ = 0, (adj A) B = 0 \Rightarrow infinite; ∣ A ∣ = 0, (adj A) B \neq = 0 \Rightarrow inconsistent$
Three-way test for a square linear system. State the case explicitly in the answer.
13.Inverse of Transpose and Product
$(A^{T})^{- 1} = (A^{- 1})^{T}, (A B)^{- 1} = B^{- 1} A^{- 1}$
Order REVERSES when you take the inverse of a product — same as transpose.
14.Symmetric and Skew-Symmetric
$A^{T} = A \Rightarrow symmetric; A^{T} = - A \Rightarrow skew-symmetric$
Every square matrix decomposes uniquely as A = ½(A + A^T) + ½(A − A^T) — symmetric plus skew-symmetric.