Matrices

2.1 Core concepts

• A matrix is an ordered rectangular array of numbers or functions; the entries are its elements, and matrices are denoted by capital letters (NCERT §3.2, p. 36).
• A matrix with m rows and n columns has order m × n and contains mn elements; the (i, j)-th element is aij, where the i-th row is [ai1 … ain] and the j-th column has entries a1j … amj (NCERT §3.2.1, p. 36).
• In this book, A = [aij]m × n and only matrices with real-number or real-valued-function entries are considered (NCERT §3.2.1, Note, p. 37).
• Types of matrices: a column matrix has only one column (order m × 1); a row matrix has only one row (order 1 × n); a square matrix has equal rows and columns (m = n); the entries a11, a22, …, ann of a square matrix form its diagonal (NCERT §3.3, pp. 39–40).
• A diagonal matrix is a square matrix with bij = 0 whenever i ≠ j; a scalar matrix is a diagonal matrix whose diagonal entries are all equal (bij = k for i = j); the identity matrix In is the scalar matrix with k = 1, so aij = 1 if i = j and 0 otherwise (NCERT §3.3 (iv)–(vi), pp. 40–41).
• Every identity matrix is a scalar matrix, but a scalar matrix is an identity matrix only when k = 1 (NCERT §3.3 (vi), p. 40). A zero (null) matrix O has all entries 0 (NCERT §3.3 (vii), p. 41).
• Two matrices are equal iff they have the same order and aij = bij for every i, j (NCERT §3.3.1 Definition 2, p. 41).
• Matrix addition: if A and B have the same order m × n, then A + B = [aij + bij]; if orders differ, A + B is not defined (NCERT §3.4.1, p. 44).
• Scalar multiplication: kA = [k aij]; negative −A = (−1)A; difference A − B = A + (−1)B (NCERT §3.4.2, pp. 45–46).
• Properties: matrix addition is commutative (A + B = B + A), associative ((A + B) + C = A + (B + C)), has additive identity O and additive inverse −A; for scalars k, l: k(A + B) = kA + kB and (k + l)A = kA + lA (NCERT §3.4.3 & §3.4.4, pp. 46–47).
• Matrix multiplication: product AB is defined iff number of columns of A equals number of rows of B; if A is m × n and B is n × p, then C = AB is m × p with cik = Σj aij bjk (NCERT §3.4.5, p. 51).
• AB defined does not imply BA defined; both AB and BA are defined together only when A is m × n and B is n × m. Even when both exist and have the same order, generally AB ≠ BA — matrix multiplication is non-commutative (NCERT §3.4.5, pp. 52–53, Examples 13 & 14).
• If AB = O for matrices A, B, it does not follow that A = O or B = O (NCERT §3.4.5, Example 15, p. 54).
• Properties of multiplication: associative (AB)C = A(BC); distributive A(B + C) = AB + AC and (A + B)C = AC + BC; existence of multiplicative identity IA = AI = A for square A (NCERT §3.4.6, p. 54).
• Transpose: if A = [aij]m × n, then A′ (or AT) = [aji]n × m — rows and columns are interchanged (NCERT §3.5 Definition 3, p. 61).
• Properties of transpose: (A′)′ = A; (kA)′ = kA′; (A + B)′ = A′ + B′; (AB)′ = B′A′ (NCERT §3.5.1, p. 61).
• A square matrix A is symmetric if A′ = A (i.e., aij = aji) and skew-symmetric if A′ = −A (i.e., aji = −aij), which forces all diagonal entries aii = 0 (NCERT §3.6 Definitions 4 & 5, pp. 63–64).
• For any square matrix A: A + A′ is symmetric and A − A′ is skew-symmetric; consequently every square matrix can be expressed as A = ½(A + A′) + ½(A − A′), a sum of a symmetric and a skew-symmetric matrix (NCERT §3.6 Theorems 1 & 2, pp. 64–65).
• A square matrix A of order m is invertible if there exists a square matrix B of order m such that AB = BA = I; B is the inverse of A, denoted A⁻¹ (NCERT §3.7 Definition 6, p. 68).
• Rectangular matrices have no inverse; if B = A⁻¹, then A = B⁻¹ (NCERT §3.7 Note, p. 69). The inverse, if it exists, is unique (Theorem 3, p. 69), and (AB)⁻¹ = B⁻¹A⁻¹ for invertible A, B of the same order (Theorem 4, p. 69).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• The Radha–Fauzia–Simran notebook/pen table on p. 35 motivates the matrix as a tabular array, and the second arrangement (rows = item type) shows how the same data can be transposed.
• The quadrilateral ABCD example on p. 37 illustrates how vertices in a plane can be packed as a 2 × 4 column-arrangement X or a 4 × 2 row-arrangement Y — a concrete row–column flip you can use to remember transpose.
• The Fatima two-factory production tables on p. 43 visualise matrix addition (corresponding entries sum) and the doubled-production table on p. 44 visualises scalar multiplication.
• The Meera–Nadeem pen/storybook computation on p. 50 walks through the row-times-column rule that defines AB; the schematic (2 × 2)(2 × 1) → (2 × 1) is the canonical conformability picture.
• Example 13 (p. 53) — a 2 × 3 times 3 × 2 product giving a 2 × 2 result and the reverse giving 3 × 3 — is the standard visual for "AB and BA need not even have the same order".

2.5 Key formulas & theorems

2.6 Solved examples (NCERT-grounded)

Example A (NCERT §3.4.5 Example 12, p. 52). A = [[6, 9], [2, 3]], B = [[2, 6, 0], [7, 9, 8]]. Find AB.

Step 1 — check conformability: 2 × 2 × 2 × 3 ⇒ AB is 2 × 3. Step 2 — compute each entry: (1,1) = 6·2 + 9·7 = 75; (1,2) = 6·6 + 9·9 = 117; (1,3) = 6·0 + 9·8 = 72; (2,1) = 2·2 + 3·7 = 25; (2,2) = 2·6 + 3·9 = 39; (2,3) = 2·0 + 3·8 = 24. Step 3 — assemble: AB = [[75, 117, 72], [25, 39, 24]].

Example B (NCERT §3.4.5 Example 14, p. 53). Show AB ≠ BA for A = [[1, 0], [0, −1]], B = [[0, 1], [1, 0]].

Step 1 — AB: [[1·0 + 0·1, 1·1 + 0·0], [0·0 + (−1)·1, 0·1 + (−1)·0]] = [[0, 1], [−1, 0]]. Step 2 — BA: [[0·1 + 1·0, 0·0 + 1·(−1)], [1·1 + 0·0, 1·0 + 0·(−1)]] = [[0, −1], [1, 0]]. Step 3 — compare: AB ≠ BA ⇒ non-commutative.

Example C (NCERT §3.4.5 Example 15, p. 54). AB = O without A = O, B = O.

Step 1 — set A = [[0, −1], [0, 2]], B = [[3, 5], [0, 0]]: both non-zero. Step 2 — compute AB: [[0·3 + (−1)·0, 0·5 + (−1)·0], [0·3 + 2·0, 0·5 + 2·0]] = [[0, 0], [0, 0]]. Step 3 — conclude: AB = O without either matrix being zero.

Example D (NCERT §3.6 Theorem 1, p. 64). Show A + A' is symmetric.

Step 1 — (A + A')' = A' + (A')': using (B + C)' = B' + C'. Step 2 — (A')' = A: so (A + A')' = A' + A. Step 3 — = A + A': hence A + A' is symmetric.

Example E (NCERT §3.7 Theorem 4, p. 69). Show (AB)⁻¹ = B⁻¹A⁻¹ for invertible A, B.

Step 1 — verify left inverse: (B⁻¹A⁻¹)(AB) = B⁻¹(A⁻¹A)B = B⁻¹IB = I. Step 2 — verify right inverse: (AB)(B⁻¹A⁻¹) = A(BB⁻¹)A⁻¹ = AIA⁻¹ = I. Step 3 — conclude: by uniqueness of the inverse, (AB)⁻¹ = B⁻¹A⁻¹.

2.4 Common confusions / NTA trap points

• "Scalar matrix vs identity matrix" — every identity is scalar, but a scalar matrix is identity only when k = 1. NTA likes to swap these in option text (NCERT §3.3 (vi), p. 40).
• "AB defined ⇒ BA defined" — false in general. Both are defined simultaneously only when A is m × n and B is n × m; in particular always defined when both are square of the same order (NCERT §3.4.5 Remark, p. 52).
• "AB ≠ BA always" — false. Diagonal matrices of the same order commute (NCERT Note, p. 53). Distractors often claim multiplication is always non-commutative.
• "AB = O ⇒ A = O or B = O" — false; the worked example with A = [[0,−1],[0,2]], B = [[3,5],[0,0]] shows AB = O with both A, B ≠ O (NCERT Example 15, p. 54).
• "Diagonal of a skew-symmetric matrix" — must be all zero. Students forget that A′ = −A forces aii = −aii ⇒ aii = 0 (NCERT §3.6 after Definition 5, p. 63).
• "(AB)′ = A′B′" — wrong; correct rule is (AB)′ = B′A′ (order reverses), and the same reversal applies to inverse: (AB)⁻¹ = B⁻¹A⁻¹ (NCERT §3.5.1, p. 61 and Theorem 4, p. 69).