Matrices

In this chapter: order of a matrix, types of matrices, equality, addition and scalar multiplication, multiplication of matrices and its properties, transpose, symmetric and skew-symmetric matrices, and invertible matrices.

Order and notation

A matrix with m rows and n columns has order m × n and contains mn elements. We write A = [a_ij]_{m × n}, where a_ij is the element in the i-th row and j-th column. Row first, column second, always.

Since a matrix with mn elements can be arranged in any order p × q with pq = mn, the number of possible orders equals the number of divisors of mn. For 24 elements the orders are 1 × 24, 2 × 12, 3 × 8, 4 × 6 and their reverses, which makes 8.

Worked example: Construct a 2 × 2 matrix A = [a_ij] with a_ij = (i + 2j)²/2.
Solution: a₁₁ = (1 + 2)²/2 = 9/2, a₁₂ = (1 + 4)²/2 = 25/2, a₂₁ = (2 + 2)²/2 = 8, a₂₂ = (2 + 4)²/2 = 18. So A = 9/225/2818.

Types of matrices

Two matrices are equal when they have the same order and every pair of corresponding elements is equal. A matrix equation therefore gives as many ordinary equations as there are elements, which is how "find x, y, z" questions are solved.

Addition and scalar multiplication

Matrices of the same order are added element by element: A + B = [a_ij + b_ij]. For a scalar k, kA = [k a_ij]. The negative is −A = (−1)A and the difference is A − B = A + (−1)B.

• Addition is commutative and associative; O is the additive identity and −A the additive inverse.
• k(A + B) = kA + kB and (k + l)A = kA + lA.

Multiplication of matrices

The product AB is defined only when the number of columns of A equals the number of rows of B. If A is m × n and B is n × p, then AB is m × p, and its (i, k) element is the sum of products of the i-th row of A with the k-th column of B:

Worked example: For the matrices in the figure, find AB and BA.
Solution: AB (2 × 2): c₁₁ = 1(2) + 2(0) + 3(1) = 5, c₁₂ = 1(1) + 2(3) + 3(−1) = 4, c₂₁ = 4(2) + 5(0) + 6(1) = 14, c₂₂ = 4(1) + 5(3) + 6(−1) = 13. So AB = 541413. BA is (3 × 2)(2 × 3) = 3 × 3: its first row is [2(1) + 1(4), 2(2) + 1(5), 2(3) + 1(6)] = [6, 9, 12], and so on. AB and BA do not even have the same order, so they certainly are not equal.

Properties of multiplication

• Associative: (AB)C = A(BC), whenever both sides are defined.
• Distributive: A(B + C) = AB + AC and (A + B)C = AC + BC.
• Identity: for a square matrix A of order n, AI_n = I_nA = A.
• Not commutative in general. AB may be defined while BA is not; even when both are defined and of the same order, they are usually different. Consequently (A + B)² = A² + AB + BA + B², which equals A² + 2AB + B² only when AB = BA.
• AB = O does not force A = O or B = O. For example, 0−102 3500 = 0000, though neither factor is zero. So you cannot cancel a matrix from both sides of an equation.

Powers are defined for square matrices: A² = AA, A³ = A²A. For patterns, compute A² and A³ and guess; for example, if A = 1101 then A² = 1201 and in general Aⁿ = 1n01, which can be proved by induction.

Transpose

The transpose A′ (or A^T) is obtained by interchanging rows and columns: if A = [a_ij] is m × n, then A′ = [a_ji] is n × m.

Symmetric and skew-symmetric matrices

• Symmetric: A′ = A, that is, a_ij = a_ji for all i, j. The matrix is a mirror image across the principal diagonal.
• Skew-symmetric: A′ = −A, that is, a_ji = −a_ij. Putting i = j gives a_ii = −a_ii, so every diagonal element of a skew-symmetric matrix is zero.
• For any square matrix A, A + A′ is symmetric and A − A′ is skew-symmetric.
• Every square matrix can be written uniquely as the sum of a symmetric and a skew-symmetric matrix: A = ½(A + A′) + ½(A − A′).

Worked example: Express A = 351−1 as the sum of a symmetric and a skew-symmetric matrix.
Solution: A′ = 315−1. P = ½(A + A′) = ½666−2 = 333−1, which is symmetric. Q = ½(A − A′) = ½04−40 = 02−20, which is skew-symmetric. Check: P + Q = 351−1 = A.

Invertible matrices

A square matrix A of order n is invertible if there is a square matrix B of the same order with AB = BA = I_n. Then B is the inverse of A, written A⁻¹. A rectangular matrix has no inverse, because AB and BA cannot both be defined and equal to the same identity matrix.

• The inverse, if it exists, is unique. (If B and C are both inverses, B = BI = B(AC) = (BA)C = IC = C.)
• (AB)⁻¹ = B⁻¹A⁻¹ for invertible A and B of the same order.

The practical method of finding A⁻¹ through the adjoint, and the condition |A| ≠ 0 for its existence, belong to the next chapter on determinants.

Common mistakes: (1) Multiplying element by element instead of row by column. (2) Expanding (A + B)² as A² + 2AB + B² without checking AB = BA. (3) Writing (AB)′ = A′B′ or (AB)⁻¹ = A⁻¹B⁻¹; both reverse the order. (4) Concluding A = O or B = O from AB = O, or cancelling A from AB = AC. (5) Forgetting that the diagonal of a skew-symmetric matrix must be all zeros, which is often the quickest way to eliminate options.

JEE and MHT‑CET focus

• Conditions for AB and BA to be defined, and the order of the product.
• Fast, accurate multiplication of 2 × 2 and 3 × 3 matrices, and powers of a matrix found from a pattern.
• Transpose rules, especially (AB)′ = B′A′, and deciding whether expressions such as AB − BA or A′BA are symmetric or skew-symmetric.
• Splitting a matrix into symmetric and skew-symmetric parts.
• Counting questions: number of possible orders, number of matrices with entries from a given set.
• Matrix equations: using equality of matrices to find unknowns, and using A² − kA + lI = O type relations to find A⁻¹ or higher powers.

Practice questions