Matrices Class 12 Notes — Quick CBSE Revision Sheet

Essential Terms & Classifications

Matrix

An ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.

Order of a Matrix

A matrix having 'm' rows and 'n' columns is called a matrix of order m x n (read as 'm by n').

Row Matrix

A matrix having only one row. Its order is of the form 1 x n.

Column Matrix

A matrix having only one column. Its order is of the form m x 1.

Square Matrix

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

Diagonal Matrix

A square matrix in which all elements except those in the principal diagonal are zero.

Scalar Matrix

A diagonal matrix in which all the diagonal elements are equal to a scalar value 'k'.

Identity Matrix

A square matrix in which elements in the main diagonal are all 1 and rest are all zero. It is denoted by 'I'.

Algebraic Operations on Matrices

Operations on matrices follow strictly defined conditions that are highly tested in CBSE Boards:

1. Addition of Matrices: Two matrices A and B can be added only if they have the same order. If A = [a_ij] and B = [b_ij] are of order m x n, then their sum A + B is defined as a matrix C = [c_ij] of order m x n where c_ij = a_ij + b_ij for all possible values of i and j.

1. Multiplication by a Scalar: If A = [a_ij] is a matrix and k is a scalar, then kA is another matrix obtained by multiplying each element of A by the scalar k. Thus, kA = [k * a_ij].

1. Multiplication of Matrices: The product of two matrices A and B is defined only if the number of columns in A is equal to the number of rows in B.

If A is an m x n matrix and B is an n x p matrix, then their product AB is a matrix C of order m x p. The element c_ij of the product matrix is obtained by multiplying the elements of the i-th row of A with corresponding elements of the j-th column of B and summing them up:
c_ij = a_i1b_1j + a_i2b_2j + ... + a_inb_nj*

Note: Matrix multiplication is not commutative in general (i.e., AB != BA, even if both products are defined).

Symmetric vs. Skew-Symmetric Matrices

Must Remember Properties & Theorems

• Properties of Transpose: (i) (A^T)^T = A; (ii) (kA)^T = k A^T (where k is a constant); (iii) (A + B)^T = A^T + B^T; (iv) (AB)^T = B^T A^T (Reversal Law).
• Any square matrix A can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix: A = (1/2)(A + A^T) + (1/2)(A - A^T). This is a highly tested 3-mark board question.
• If A and B are symmetric matrices of the same order, then AB + BA is symmetric, while AB - BA is skew-symmetric.
• Matrix multiplication is distributive over matrix addition: A(B + C) = AB + AC.
• If AB = O (where O is the zero matrix), it is NOT necessary that either A = O or B = O. Product of two non-zero matrices can be a zero matrix.
• For any square matrix A of order n, AI = IA = A, where I is the identity matrix of order n.

Quick Worked Examples

• {"title":"Example 1: Matrix Equation Solving","content":"Find the values of x and y if: \n2 [[x, 5], [7, y - 3]] + [[3, -4], [1, 2]] = [[7, 6], [15, 14]]\n\nSolution:\n1. Multiply the scalar 2 into the first matrix:\n[[2x, 10], [14, 2y - 6]] + [[3, -4], [1, 2]] = [[7, 6], [15, 14]]\n2. Add the two LHS matrices element-wise:\n[[2x + 3, 6], [15, 2y - 4]] = [[7, 6], [15, 14]]\n3. Equate the corresponding elements:\n2x + 3 = 7 => 2x = 4 => x = 2\n2y - 4 = 14 => 2y = 18 => y = 9\nAnswer: x = 2, y = 9.*"}
• {"title":"Example 2: Symmetric and Skew-Symmetric Decomposition","content":"Express the matrix A = [[2, 4], [5, 6]] as the sum of a symmetric and a skew-symmetric matrix.\n\nSolution:\n1. Find A^T:\nA^T = [[2, 5], [4, 6]]\n2. Calculate Symmetric part P = (1/2)(A + A^T):\nP = (1/2) ([[2+2, 4+5], [5+4, 6+6]]) = (1/2) [[4, 9], [9, 12]] = [[2, 4.5], [4.5, 6]]\n3. Calculate Skew-symmetric part Q = (1/2)(A - A^T):\nQ = (1/2) ([[2-2, 4-5], [5-4, 6-6]]) = (1/2) [[0, -1], [1, 0]] = [[0, -0.5], [0.5, 0]]\n4. Verify: P + Q = [[2+0, 4.5-0.5], [4.5+0.5, 6+0]] = [[2, 4], [5, 6]] = A. (Verified successfully)."}

Board Exam Trap Alerts

• The Reversal Law Trap: In exams, students often write (AB)^T = A^T B^T. Remember, it is always B^T A^T. This reversal rule also applies to matrix inverses: (AB)^-1 = B^-1 * A^-1.
• Properties of Diagonal Elements: If a question asks you to write a skew-symmetric matrix, remember to write diagonal elements as exact zeros first, otherwise you will lose marks instantly.
• Order Mismatch in Multiplication: Before attempting matrix multiplication, always explicitly write down the order of both matrices (e.g., 3x2 and 2x3) to confirm if the inner numbers match, determining if the operation is even defined.

Quick Revision Check

• If a matrix has 24 elements, what are all the possible orders it can have? The possible orders are the ordered pairs of natural numbers whose product is 24. These are: 1x24, 24x1, 2x12, 12x2, 3x8, 8x3, 4x6, and 6x4. Thus, there are 8 possible orders.
• Prove that the diagonal elements of a skew-symmetric matrix are always zero. Let A = [a_ij] be a skew-symmetric matrix. By definition, a_ij = -a_ji for all i, j. For the diagonal elements, i = j. Substituting this gives: a_ii = -a_ii => 2 * a_ii = 0 => a_ii = 0. Hence, all diagonal elements of a skew-symmetric matrix are zero.
• If A is a square matrix such that A^2 = A, then what is the value of (I + A)^3 - 7A? Expand using binomial expansion: (I + A)^3 = I^3 + 3I^2A + 3IA^2 + A^3. Since I^n = I, IA = A, and A^2 = A: A^3 = A^2 A = A A = A^2 = A. Therefore, (I + A)^3 = I + 3A + 3A + A = I + 7A. Substituting this back: (I + A)^3 - 7A = (I + 7A) - 7A = I. Answer is I.
• If A and B are symmetric matrices of the same order, is AB - BA symmetric or skew-symmetric? Let Y = AB - BA. Take transpose on both sides: Y^T = (AB - BA)^T = (AB)^T - (BA)^T = (B^T A^T) - (A^T B^T). Since A and B are symmetric, A^T = A and B^T = B. Thus, Y^T = BA - AB = -(AB - BA) = -Y. Since Y^T = -Y, the matrix AB - BA is a skew-symmetric matrix.

Frequently Asked Questions