CBSE Class 12 Maths: Determinants (Exercise 4.1)

What is a Determinant?

In simple terms, a determinant is a special scalar value that can be computed from the elements of a square matrix. It's a numerical property that reveals important characteristics of the matrix it's associated with. Unlike a matrix itself, which is an arrangement of numbers, a determinant is a single number. For a matrix A, its determinant is denoted by det(A) or by enclosing the matrix elements within vertical bars, like |A| or |a b / c d|. It's important to remember that determinants are defined only for square matrices (matrices with an equal number of rows and columns).

Why are determinants important? They help us determine if a system of linear equations has a unique solution, if a matrix is invertible, and they appear in formulas for finding the area of a triangle given its vertices or the volume of a tetrahedron. The value of a determinant can also be thought of as a scaling factor for area or volume when a linear transformation is applied. A non-zero determinant signifies that the transformation expands or contracts the space, while a zero determinant implies that the transformation collapses the space into a lower dimension, making the matrix singular (non-invertible).

Determinant of a 2x2 Matrix

Calculating the determinant of a 2x2 matrix is straightforward. If you have a matrix A given by:

A = [[a, b], [c, d]]

The determinant of A, det(A) or |A|, is calculated by subtracting the product of the elements on the anti-diagonal (c and b) from the product of the elements on the main diagonal (a and d).

Formula:
|A| = ad - bc

Let's consider a quick example:
If A = [[2, 3], [4, 5]]
Then |A| = (2 5) - (3 4) = 10 - 12 = -2.

This formula is fundamental and will be used repeatedly, even when calculating larger 3x3 determinants. Ensure you remember the order of subtraction: main diagonal product minus anti-diagonal product. A common mistake is to reverse the order, leading to an incorrect sign for the determinant. Always double-check your calculations.

Calculating the Determinant of a 3x3 Matrix

1. Step 1: Choose a Row or Column — To find the determinant of a 3x3 matrix, you can expand it along any row or any column. The final value will be the same regardless of your choice. It's often strategic to choose a row or column that contains the most zeros, as this simplifies calculations.
2. Step 2: Apply the Sign Convention — Each element in the matrix has an associated sign based on its position (i, j). The sign is determined by (-1)^(i+j). For a 3x3 matrix, the signs are arranged as: [[+, -, +], [-, +, -], [+, -, +]] When expanding, you multiply each element by its sign and its minor (a 2x2 determinant).
3. Step 3: Calculate the Minors — For each element chosen in your selected row/column, its minor is the determinant of the 2x2 matrix formed by deleting the row and column containing that element. For example, if you choose element a_11, its minor M_11 is the determinant of the 2x2 matrix remaining after removing the 1st row and 1st column.
4. Step 4: Compute the Cofactors — The cofactor C_ij of an element a_ij is (-1)^(i+j) * M_ij, where M_ij is the minor of a_ij. Essentially, it's the minor multiplied by its sign from Step 2.
5. Step 5: Sum the Products — The determinant of the 3x3 matrix is the sum of the products of each element in the chosen row/column with its corresponding cofactor. For expansion along the first row (elements a_11, a_12, a_13): |A| = a_11 C_11 + a_12 C_12 + a_13 C_13 Or, in terms of minors: |A| = a_11 M_11 - a_12 M_12 + a_13 M_13 (using the sign convention directly).

Worked Examples

• Example 1: Find the determinant of the matrix A = [[2, -1], [3, 4]]. Step 1: Identify the elements a, b, c, d. Here, a = 2, b = -1, c = 3, d = 4. Step 2: Apply the formula for a 2x2 determinant: |A| = ad - bc. |A| = (2 4) - (-1 3) Step 3: Calculate the products and subtract. |A| = 8 - (-3) |A| = 8 + 3 Final answer: |A| = 11
• Example 2: Evaluate the determinant of the matrix B = [[1, 2, 3], [4, 1, 2], [0, 5, 1]]. Step 1: Choose a row or column for expansion. Let's choose the first row (R1) because it's a common practice, but observe that C1 has a zero, which could simplify calculations. Step 2: Apply the sign convention for R1: (+, -, +). Step 3: Calculate the minors and cofactors for elements in R1. For b_11 = 1: Minor M_11 = |[1, 2], [5, 1]| = (11) - (25) = 1 - 10 = -9. Cofactor C_11 = +1 (-9) = -9. For b_12 = 2: Minor M_12 = |[4, 2], [0, 1]| = (41) - (20) = 4 - 0 = 4. Cofactor C_12 = -1 (4) = -4. For b_13 = 3: Minor M_13 = |[4, 1], [0, 5]| = (45) - (10) = 20 - 0 = 20. Cofactor C_13 = +1 (20) = 20. Step 4: Sum the products of elements and their cofactors. |B| = b_11 C_11 + b_12 C_12 + b_13 C_13 |B| = 1 (-9) + 2 (-4) + 3 * (20) |B| = -9 - 8 + 60 |B| = -17 + 60 Final answer: |B| = 43
• Example 3: If |[[x, 2], [18, x]]| = |[[6, 2], [18, 6]]|, find the value(s) of x. Step 1: Calculate the determinant of the left-hand side (LHS) matrix. LHS = (x x) - (2 18) = x^2 - 36 Step 2: Calculate the determinant of the right-hand side (RHS) matrix. RHS = (6 6) - (2 18) = 36 - 36 = 0 Step 3: Equate the two determinants and solve for x. x^2 - 36 = 0 x^2 = 36 Take the square root of both sides: x = ±√36 Final answer: x = ±6

Exam Tips & Common Mistakes to Avoid

When solving problems related to Determinants Ex 4.1, especially under exam pressure, students often make a few recurring mistakes. Being aware of these can help you avoid losing valuable marks:

1. Sign Errors in 3x3 Expansion: This is perhaps the most common mistake. Always remember the alternating sign pattern (+ - +) when expanding a 3x3 determinant. It's a_11M_11 - a_12M_12 + a_13M_13 (if expanding along the first row), NOT a_11M_11 + a_12M_12 + a_13M_13. Double-check the (-1)^(i+j) rule for each position.
2. Incorrect 2x2 Calculation: For a 2x2 matrix [[a, b], [c, d]], the determinant is ad - bc. Many students sometimes reverse the subtraction, writing bc - ad. This will give you the correct magnitude but the wrong sign. Always remember 'main diagonal minus anti-diagonal'.
3. Calculation Mistakes: Simple arithmetic errors (addition, subtraction, multiplication) are common. Take your time, especially with negative numbers. Performing a quick mental check or re-calculation can save you.
4. Not Choosing the Easiest Row/Column: While any row or column works, choosing one with zeros significantly reduces calculation. If a row/column has two zeros, your work becomes incredibly simple, often reducing the problem to a single 2x2 determinant calculation.

Practice Questions with Solutions

• Q: Evaluate the determinant: |[-1, 2], [3, -4]]| A: Step 1: Identify the elements: a = -1, b = 2, c = 3, d = -4. Step 2: Apply the 2x2 determinant formula: ad - bc. Determinant = (-1 -4) - (2 3) Step 3: Perform the multiplication and subtraction. Determinant = 4 - 6 Final answer: -2
• Q: Find the value of x if |[[x, 3], [4, 5]]| = 7. A: Step 1: Calculate the determinant of the given 2x2 matrix. Determinant = (x 5) - (3 4) = 5x - 12 Step 2: Equate the determinant to 7. 5x - 12 = 7 Step 3: Solve the linear equation for x. 5x = 7 + 12 5x = 19 x = 19/5 Final answer: x = 19/5
• Q: Evaluate the determinant: |[[3, -4, 5], [1, 1, -2], [2, 3, 1]]| A: Step 1: Choose to expand along the first row (R1). Step 2: Apply the sign convention for R1: (+, -, +). Step 3: Calculate the cofactors for each element in R1. For a_11 = 3: Cofactor C_11 = +1 |[1, -2], [3, 1]| = (11) - (-23) = 1 - (-6) = 1 + 6 = 7. For a_12 = -4: Cofactor C_12 = -1 |[1, -2], [2, 1]| = -1 ((11) - (-22)) = -1 (1 - (-4)) = -1 (1 + 4) = -1 5 = -5. For a_13 = 5: Cofactor C_13 = +1 |[1, 1], [2, 3]| = (13) - (12) = 3 - 2 = 1. Step 4: Sum the products of elements and their cofactors. Determinant = 3 C_11 + (-4) C_12 + 5 C_13 Determinant = 3 7 + (-4) (-5) + 5 * 1 Determinant = 21 + 20 + 5 Final answer: 46
• Q: If A = [[cos(theta), -sin(theta)], [sin(theta), cos(theta)]], find |A|. A: Step 1: Identify the elements: a = cos(theta), b = -sin(theta), c = sin(theta), d = cos(theta). Step 2: Apply the 2x2 determinant formula: ad - bc. |A| = (cos(theta) cos(theta)) - (-sin(theta) sin(theta)) Step 3: Simplify the expression. |A| = cos^2(theta) - (-sin^2(theta)) |A| = cos^2(theta) + sin^2(theta) Step 4: Use the trigonometric identity cos^2(theta) + sin^2(theta) = 1. Final answer: 1

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