Matrices Exercise 3.1 for CBSE Class 12 Maths: A Deep Dive

Understanding Matrices: Definition, Order, and Elements

A matrix is simply a rectangular arrangement of numbers or functions, called elements or entries, arranged in rows and columns. Think of it like a table, but with specific mathematical rules attached. For example, if you have data like marks of students in different subjects, a matrix can neatly represent this.

The order of a matrix is defined by the number of rows and the number of columns it has. If a matrix has m rows and n columns, its order is denoted as m × n (read as 'm by n'). The number of elements in such a matrix will be m × n. For instance, a matrix with 3 rows and 2 columns is a 3 × 2 matrix, containing 3 * 2 = 6 elements.

Each element in a matrix is uniquely identified by its position. We denote the element in the i-th row and j-th column as a_ij. The first subscript i always refers to the row number, and the second subscript j refers to the column number. So, a_12 would be the element in the first row and second column, while a_21 would be the element in the second row and first column. This notation is crucial for accurately describing and working with matrices. Understanding these foundational concepts is key to solving problems in Exercise 3.1.

Key Definitions in Matrices

Matrix

A rectangular array of numbers or functions, arranged in rows and columns. These numbers or functions are called the elements or entries of the matrix.

Order of a Matrix

If a matrix has 'm' rows and 'n' columns, its order is 'm × n' (read as 'm by n'). The total number of elements in such a matrix is m multiplied by n.

Elements of a Matrix

The numbers or functions that constitute the matrix. An element is denoted by a_ij, where i represents its row number and j represents its column number.

Equality of Matrices

Two matrices, A and B, are said to be equal if and only if: 1. They are of the same order. 2. Each element of A is equal to the corresponding element of B (i.e., a_ij = b_ij for all i and j).

Worked Examples from Matrices Ex 3.1

• Example 1: Construct a 2 × 2 matrix A = [a_ij] whose elements are given by a_ij = (i + j)^2 / 2. Step 1: Understand the order. We need to construct a 2 × 2 matrix, meaning it will have 2 rows and 2 columns. The elements will be a_11, a_12, a_21, a_22. Step 2: Calculate each element using the given formula. For a_11: i=1, j=1. So, a_11 = (1 + 1)^2 / 2 = (2)^2 / 2 = 4 / 2 = 2. For a_12: i=1, j=2. So, a_12 = (1 + 2)^2 / 2 = (3)^2 / 2 = 9 / 2. For a_21: i=2, j=1. So, a_21 = (2 + 1)^2 / 2 = (3)^2 / 2 = 9 / 2. For a_22: i=2, j=2. So, a_22 = (2 + 2)^2 / 2 = (4)^2 / 2 = 16 / 2 = 8. Step 3: Assemble the matrix. The matrix A is: `` A = [[2, 9/2], [9/2, 8]] ``
• Example 2: If a matrix has 8 elements, what are the possible orders it can have? Step 1: Understand the relationship between elements and order. The number of elements in a matrix of order m × n is m × n. We are given that m × n = 8. Step 2: Find all pairs of positive integers (m, n) whose product is 8. The possible pairs of (m, n) are the factors of 8: (1, 8) – This means 1 row and 8 columns. (8, 1) – This means 8 rows and 1 column. (2, 4) – This means 2 rows and 4 columns. (4, 2) – This means 4 rows and 2 columns. Step 3: List the possible orders. The possible orders are 1 × 8, 8 × 1, 2 × 4, and 4 × 2.

Exam Tips & Common Mistakes in Matrices Ex 3.1

When tackling problems from Exercise 3.1, keep these points in mind to avoid common errors and score full marks:

1. Don't Confuse a_ij and a_ji: Always remember that the first subscript i is for the row number, and the second j is for the column number. a_12 is not the same as a_21 unless the elements themselves are equal. This is a very common mistake, especially when constructing matrices.
2. Order Matters for Equality: For two matrices to be equal, they must first have the exact same order. A 2 × 3 matrix can never be equal to a 3 × 2 matrix, regardless of their elements.
3. Corresponding Elements: When equating two matrices, ensure you equate corresponding elements. a_11 of the first matrix equals a_11 of the second matrix, and so on. Do not mix up the positions.
4. Solving Systems of Equations: Problems involving matrix equality often reduce to solving a system of linear equations. Double-check your algebraic manipulations (addition, subtraction, substitution) to find the correct values of unknowns.

Practice Questions with Solutions

• Q: Construct a 3 × 2 matrix A = [a_ij] where a_ij = |i - 2j|. A: Step 1: Identify the elements to be calculated. For a 3 × 2 matrix, the elements are a_11, a_12, a_21, a_22, a_31, a_32. Step 2: Calculate each element using the formula a_ij = |i - 2j|. a_11 = |1 - 2(1)| = |-1| = 1 a_12 = |1 - 2(2)| = |-3| = 3 a_21 = |2 - 2(1)| = |0| = 0 a_22 = |2 - 2(2)| = |-2| = 2 a_31 = |3 - 2(1)| = |1| = 1 a_32 = |3 - 2(2)| = |-1| = 1 Step 3: Form the matrix. `` A = [[1, 3], [0, 2], [1, 1]] `` Final answer: The constructed matrix is A = [[1, 3], [0, 2], [1, 1]].
• Q: If a matrix has 13 elements, what are the possible orders it can have? A: Step 1: Understand that the product of rows (m) and columns (n) equals the number of elements. Given, number of elements = 13. So, m × n = 13. Step 2: Find pairs of positive integers whose product is 13. Since 13 is a prime number, its only positive integer factors are 1 and 13. Possible pairs (m, n) are (1, 13) and (13, 1). Step 3: List the possible orders. The possible orders are 1 × 13 and 13 × 1. Final answer: The possible orders are 1 × 13 and 13 × 1.
• Q: Find the values of x, y, and z from the following matrix equality: `` [[x+y, 2], [5+z, xy]] = [[6, 2], [5, 8]] `` A: Step 1: Understand that for matrices to be equal, their corresponding elements must be equal. From the given equality, we can form a system of equations. Step 2: Equate corresponding elements. x + y = 6 (Equation 1) 2 = 2 (This is consistent) 5 + z = 5 (Equation 2) xy = 8 (Equation 3) Step 3: Solve the system of equations. From Equation 2: 5 + z = 5 => z = 5 - 5 => z = 0. From Equation 1, express y in terms of x: y = 6 - x. Substitute y into Equation 3: x(6 - x) = 8. 6x - x^2 = 8 x^2 - 6x + 8 = 0 Factor the quadratic equation: (x - 2)(x - 4) = 0. So, x = 2 or x = 4. Step 4: Find the corresponding y values. If x = 2, then y = 6 - 2 = 4. If x = 4, then y = 6 - 4 = 2. Final answer: The values are x = 2, y = 4, z = 0 OR x = 4, y = 2, z = 0.
• Q: For the matrix A = [[2, 5, 19, -7], [35, -2, 5/2, 12], [root(3), 1, -5, 17]], Write: (i) The order of the matrix. (ii) The number of elements. (iii) Write the elements a_13, a_21, a_33, a_24, a_23. A: Step 1: Determine the number of rows and columns. There are 3 rows and 4 columns. (i) The order of the matrix is 3 × 4. Step 2: Calculate the total number of elements. Number of elements = (number of rows) × (number of columns) = 3 × 4 = 12. (ii) The number of elements is 12. Step 3: Identify the elements at the specified positions. a_13 (1st row, 3rd column) = 19 a_21 (2nd row, 1st column) = 35 a_33 (3rd row, 3rd column) = -5 a_24 (2nd row, 4th column) = 12 a_23 (2nd row, 3rd column) = 5/2 Final answer: (i) 3 × 4; (ii) 12; (iii) a_13 = 19, a_21 = 35, a_33 = -5, a_24 = 12, a_23 = 5/2.

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