xSkill: notes, lectures and practice for CBSE and CSE

1.1 Order, entries and types A matrix is a rectangular arrangement of numbers in rows and columns. A matrix with m rows and n columns has order m by n and mn entries. Two matrices can be equal only when they have the same order and every matching entry is equal. This equality rule is the basis of most numerical questions on matrices in the board paper. The type families are worth naming exactly. A row matrix has one row, a column matrix has one column, a rectangular matrix has unequal numbers of rows and columns, and a square matrix has the same number of rows and columns. Inside square matrices come diagonal (all non-diagonal entries zero), scalar (a diagonal matrix with equal diagonal entries), identity (scalar with ones on the diagonal) and zero (every entry zero) matrices.

A = [a_{ij}]_{m \times n}, (A)_{ij} = a_{ij}

General matrix and its entry notation

1.2 Addition, scalar multiplication and products Addition is defined only for matrices of the same order, and it works entrywise. Scalar multiplication multiplies every entry by the number. Both operations are commutative and associative for matching orders, and they distribute over each other. Matrix multiplication is the one operation that needs care. The product AB exists only when the number of columns of A equals the number of rows of B. If A has order m by n and B has order n by p, then AB has order m by p, and the entry in row i and column j comes from multiplying row i of A with column j of B and adding the results. The order rule alone answers many one mark questions.

(A B)_{ij} = k = 1 \sum n a_{ik} b_{k j}

Entry formula for the product of two matrices

(A B)_{m \times p} = A_{m \times n} B_{n \times p}

The product is defined only when the inner orders match

A + B = B + A, (A B) C = A (B C), A (B + C) = A B + A C

Arithmetic laws, but AB need not equal BA

1.3 Transpose, symmetry and inverses The transpose swaps rows with columns. Transposing twice brings back the original matrix, transposing a sum transposes each part, and transposing a product reverses the order. A square matrix is symmetric when it equals its own transpose, and skew symmetric when it equals the negative of its transpose. Every square matrix splits in exactly one way into a symmetric part plus a skew symmetric part, using half of the sum and half of the difference with its transpose. A square matrix is invertible when another matrix B satisfies AB = BA = I. Only non-singular matrices have inverses, and the inverse of a product reverses the order.

(A^{'})^{'} = A, (A + B)^{'} = A^{'} + B^{'}, (k A)^{'} = k A^{'}, (A B)^{'} = B^{'} A^{'}

Transpose rules, with the reversed order in the product

A = \frac{1}{2} (A + A^{'}) + \frac{1}{2} (A - A^{'})

Every square matrix as symmetric plus skew symmetric part

(A B)^{- 1} = B^{- 1} A^{- 1}, (A^{'})^{- 1} = (A^{- 1})^{'}

Inverse rules for products and transposes

Exam

Before multiplying, write the orders under both matrices and check that the inner numbers match. Before adding, check that both orders are identical. Most lost marks in this chapter come from skipping that check, not from arithmetic.