Matrices Class 12 Maths: Chapter 3 Notes

Key Definitions in Matrices

Matrix

A rectangular array or arrangement of numbers or functions, arranged in rows and columns. Represented as A = [a_ij]_m×n.

Order of a Matrix

A matrix having 'm' rows and 'n' columns is called a matrix of order m × n.

Square Matrix

A matrix in which the number of rows is equal to the number of columns (m = n).

Identity Matrix (I)

A square matrix where all diagonal elements are 1 and all non-diagonal elements are 0. It is the multiplicative identity, i.e., AI = IA = A.

Zero or Null Matrix

A matrix in which all elements are zero. It is the additive identity.

Transpose of a Matrix (A' or A^T)

The matrix obtained by interchanging the rows and columns of a given matrix A.

Symmetric Matrix

A square matrix A for which A' = A. This means a_ij = a_ji for all i, j.

Skew-Symmetric Matrix

A square matrix A for which A' = -A. This means a_ij = -a_ji for all i, j, and all diagonal elements are zero.

Invertible Matrix

A square matrix A is invertible if there exists a square matrix B such that AB = BA = I. The matrix B is called the inverse of A, denoted by A⁻¹.

Operations on Matrices

Understanding matrix operations is fundamental. There are four main operations: addition, subtraction, scalar multiplication, and matrix multiplication.

1. Addition and Subtraction: These operations are only possible between matrices of the same order. If A = [a_ij] and B = [b_ij] are two matrices of order m × n, then their sum A + B is a matrix C = [c_ij] of the same order, where each element c_ij = a_ij + b_ij. Subtraction follows the same principle: A - B results in a matrix where each element is a_ij - b_ij.

2. Scalar Multiplication: To multiply a matrix A by a scalar (a real number) 'k', you simply multiply every element of the matrix by k. So, kA = k[a_ij] = [ka_ij]. For example, if you multiply a 2x2 matrix by 3, all four elements inside the matrix get multiplied by 3.

3. Matrix Multiplication: This is the most complex operation. The product of two matrices A and B, denoted as AB, is defined only if the number of columns in matrix A is equal to the number of rows in matrix B. If A is an m × n matrix and B is an n × p matrix, their product AB will be an m × p matrix. To find the element in the i-th row and j-th column of AB, you take the dot product of the i-th row of A and the j-th column of B. This process is not commutative, meaning AB ≠ BA in most cases.

Must Remember: Key Properties and Formulas

• Non-Commutativity of Multiplication: For matrices A and B, in general, AB ≠ BA.
• Associative Law: Matrix addition and multiplication are associative: (A + B) + C = A + (B + C) and (AB)C = A(BC).
• Distributive Law: A(B + C) = AB + AC and (A + B)C = AC + BC.
• Transpose Properties: (A')' = A, (kA)' = kA', (A + B)' = A' + B', and the reversal law for products: (AB)' = B'A'.
• Symmetric and Skew-Symmetric Sum: Any square matrix A can be expressed as the sum of a symmetric and a skew-symmetric matrix: A = ½(A + A') + ½(A - A').
• Condition for Inverse: A square matrix A is invertible if and only if its determinant is non-zero (det(A) ≠ 0). Such a matrix is called a non-singular matrix.
• Inverse Formula: The inverse of a non-singular square matrix A is given by A⁻¹ = (1/det(A)) * adj(A), where adj(A) is the adjoint of matrix A.
• Inverse of a Product: For invertible matrices A and B of the same order, (AB)⁻¹ = B⁻¹A⁻¹.
• Solving Linear Equations: A system of linear equations AX = B can be solved using matrices as X = A⁻¹B, provided A is invertible.

Finding the Inverse using Elementary Operations

1. —
2. —
3. —
4. —

Worked Mini-Examples

• {"id":"1","title":"Matrix Multiplication","bodyMarkdown":"If A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]], find AB.\n\nSolution:\nAB = [[(10 + 21), (11 + 20)], [(30 + 41), (31 + 40)]]\nAB = [[2, 1], [4, 3]]"}
• {"id":"2","title":"Finding the Inverse of a 2x2 Matrix","bodyMarkdown":"Find the inverse of A = [[2, 5], [1, 3]].\n\nSolution:\n1. Find determinant: det(A) = (2)(3) - (5)(1) = 6 - 5 = 1. Since det(A) ≠ 0, the inverse exists.\n2. Find adjoint: For a 2x2 matrix [[a, b], [c, d]], adj(A) is [[d, -b], [-c, a]]. So, adj(A) = [[3, -5], [-1, 2]].\n3. Calculate inverse: A⁻¹ = (1/det(A)) adj(A) = (1/1) [[3, -5], [-1, 2]] = [[3, -5], [-1, 2]]."}

Common Exam Traps

Trap: Assuming matrix multiplication is commutative (AB = BA). This is almost always false and a common mistake. Always calculate AB and BA separately if asked.

Marking Cue: Before finding the inverse A⁻¹, always calculate the determinant of A first. Explicitly state det(A) ≠ 0, therefore the matrix is non-singular and the inverse exists. This shows the examiner you understand the prerequisite condition and can fetch you a mark even if you make a calculation error later.

Trap: Forgetting the reversal law for transpose and inverse of a product. Remember, (AB)' = B'A' and (AB)⁻¹ = B⁻¹A⁻¹. The order is reversed!

Practice Questions with Solutions

• If a matrix has 13 elements, what are the possible orders it can have? Since 13 is a prime number, the only possible orders are 1 × 13 and 13 × 1.
• For any square matrix A, what type of matrix is A - A'? A - A' is always a skew-symmetric matrix because (A - A')' = A' - (A')' = A' - A = -(A - A').
• If A is a 3 × 2 matrix and B is a 2 × 4 matrix, what is the order of the product AB? The order of AB is 3 × 4.
• What is the value of A⁻¹A, where A is an invertible matrix? By definition, A⁻¹A = I, the identity matrix of the same order as A.

Frequently Asked Questions

Frequently Asked Questions