Matrices

A matrix is a rectangular array of numbers (or functions) arranged in horizontal rows and vertical columns, enclosed in brackets. The numbers are called the elements or entries of the matrix. Matrices are usually named with capital letters A, B, C, ... and their elements with lower-case letters.

If a matrix A has m rows and n columns, we say A is a matrix of order m × n (read as "m by n"). The general element in the i-th row and j-th column is written a_ij, and the whole matrix is written compactly as A = [a_ij]_m×n, where 1 ≤ i ≤ m and 1 ≤ j ≤ n. A matrix of order m × n has exactly m×n elements.

Types of matrices commonly used:

• Row matrix: exactly one row (order 1×n).
• Column matrix: exactly one column (order m×1).
• Square matrix: number of rows = number of columns (order n×n). The elements a₁₁, a₂₂, ..., a_nn form the principal diagonal.
• Diagonal matrix: a square matrix in which every non-diagonal element is 0 (diagonal entries may be anything, including 0).
• Scalar matrix: a diagonal matrix in which all diagonal elements are equal.
• Identity (unit) matrix I_n: a scalar matrix in which every diagonal element equals 1.
• Zero (null) matrix: every element is 0, denoted O.

Worked Example. Construct a 2×2 matrix A = [a_ij] whose elements are given by a_ij = (i + 2j)²⁄2.

Here A has order 2×2, so i = 1, 2 and j = 1, 2. Substituting:

• a₁₁ = (1+2)²/2 = 9/2
• a₁₂ = (1+4)²/2 = 25/2
• a₂₁ = (2+2)²/2 = 16/2 = 8
• a₂₂ = (2+4)²/2 = 36/2 = 18

So A = [[9/2, 25/2], [8, 18]].

Two matrices A = [a_ij] and B = [b_ij] are said to be equal, written A = B, if and only if (i) they have the same order, and (ii) every corresponding pair of elements is equal, i.e. a_ij = b_ij for all i, j. Equality of matrices is often used to set up simultaneous equations to find unknown entries.

Addition and subtraction are defined only when the two matrices have the same order. If A = [a_ij]_m×n and B = [b_ij]_m×n, then A + B = [a_ij + b_ij]_m×n, i.e. corresponding elements are added. Subtraction A − B is defined as A + (−1)B, subtracting corresponding elements.

Matrix addition satisfies: commutativity (A+B = B+A), associativity ((A+B)+C = A+(B+C)), existence of an additive identity (A + O = A), and existence of an additive inverse (A + (−A) = O).

Scalar multiplication: if k is a number (scalar) and A = [a_ij]_m×n, then kA = [k·a_ij]_m×n — every entry is multiplied by k. This satisfies k(A+B) = kA+kB and (k+l)A = kA+lA for scalars k, l.

Worked Example. If A = [[2, 3], [1, 4]] and B = [[1, −1], [0, 2]], find 2A − 3B.

2A = [[4, 6], [2, 8]] and 3B = [[3, −3], [0, 6]].

2A − 3B = [[4−3, 6−(−3)], [2−0, 8−6]] = [[1, 9], [2, 2]].

The product AB of two matrices A and B is defined only when the number of columns of A equals the number of rows of B. If A is of order m×n and B is of order n×p, then AB exists and is of order m×p. The element in the i-th row and j-th column of AB is obtained by multiplying corresponding elements of the i-th row of A with the j-th column of B and adding the results: (AB)_ij = Σ_k=1ⁿ a_ik b_kj.

A crucial fact for CBSE and competitive exams: matrix multiplication is not commutative in general, i.e. AB ≠ BA even when both products are defined and of the same order. Matrix multiplication is, however, associative — (AB)C = A(BC) — and distributive over addition — A(B+C) = AB+AC and (A+B)C = AC+BC. Multiplication by the identity matrix leaves a matrix unchanged: AI = IA = A. Also, the product of two non-zero matrices can be a zero matrix, unlike ordinary numbers.

Worked Example. Let A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 2]]. Find AB and BA, and verify AB ≠ BA.

AB: row1·col1 = 1(2)+2(1) = 4; row1·col2 = 1(0)+2(2) = 4; row2·col1 = 3(2)+4(1) = 10; row2·col2 = 3(0)+4(2) = 8. So AB = [[4, 4], [10, 8]].

BA: row1·col1 = 2(1)+0(3) = 2; row1·col2 = 2(2)+0(4) = 4; row2·col1 = 1(1)+2(3) = 7; row2·col2 = 1(2)+2(4) = 10. So BA = [[2, 4], [7, 10]].

Since AB = [[4,4],[10,8]] and BA = [[2,4],[7,10]] are different, AB ≠ BA — confirming multiplication is not commutative.

The transpose of a matrix A of order m×n, denoted A′ (or A^T), is the matrix of order n×m obtained by interchanging the rows and columns of A, so that (A′)_ij = a_ji.

Properties of transpose (for matrices of suitable orders and scalar k):

• (A′)′ = A
• (kA)′ = kA′
• (A + B)′ = A′ + B′
• (AB)′ = B′A′ (note the reversal of order)

A square matrix A is symmetric if A′ = A, which means a_ij = a_ji for all i, j (the matrix is a mirror image of itself about the principal diagonal). A square matrix A is skew-symmetric if A′ = −A, which means a_ij = −a_ji for all i, j; setting i = j gives a_ii = −a_ii, so every diagonal element of a skew-symmetric matrix must be 0.

An important theorem: for any square matrix A, ½(A + A′) is always symmetric and ½(A − A′) is always skew-symmetric, and their sum reconstructs A. This means every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

Worked Example. Express A = [[3, 5], [1, −1]] as the sum of a symmetric and a skew-symmetric matrix.

A′ = [[3, 1], [5, −1]].

A + A′ = [[6, 6], [6, −2]], so P = ½(A + A′) = [[3, 3], [3, −1]] — check: P′ = P, so P is symmetric.

A − A′ = [[0, 4], [−4, 0]], so Q = ½(A − A′) = [[0, 2], [−2, 0]] — check: Q′ = −Q, so Q is skew-symmetric.

P + Q = [[3, 3], [3, −1]] + [[0, 2], [−2, 0]] = [[3, 5], [1, −1]] = A, as required.

An elementary operation (transformation) on a matrix is one of the following, applied to rows (or, similarly, to columns):

1. Interchange of any two rows: R_i ↔ R_j
2. Multiplication of the elements of a row by a non-zero scalar: R_i → kR_i (k ≠ 0)
3. Addition to the elements of a row, the corresponding elements of another row multiplied by a non-zero scalar: R_i → R_i + kR_j

A square matrix A is called invertible if there exists a square matrix B of the same order such that AB = BA = I. The matrix B is called the inverse of A, written A⁻¹. The inverse, if it exists, is always unique.

Finding the inverse by elementary row operations: Write A = IA. Apply a sequence of elementary row operations to the matrix on the left (A) to reduce it to the identity matrix I, applying the exact same operations simultaneously to the I on the right-hand side. When the left side becomes I, the matrix that has appeared on the right side is A⁻¹. (If, at any stage, all entries of an entire row on the left become zero, A has no inverse.) An analogous method using column operations starts from A = AI.

Worked Example. Find A⁻¹ for A = [[2, 1], [1, 1]] using elementary row operations.

Write A = IA: [[2, 1], [1, 1]] = [[1, 0], [0, 1]] A.

Apply R₁ → R₁ − R₂: [[1, 0], [1, 1]] = [[1, −1], [0, 1]] A.

Apply R₂ → R₂ − R₁: [[1, 0], [0, 1]] = [[1, −1], [−1, 2]] A.

The left side is now I, so A⁻¹ = [[1, −1], [−1, 2]].

Check: A·A⁻¹ = [[2,1],[1,1]]·[[1,−1],[−1,2]] = [[2(1)+1(−1), 2(−1)+1(2)], [1(1)+1(−1), 1(−1)+1(2)]] = [[1, 0], [0, 1]] = I.