Problem 11 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

EVALUATING DETERMINANTS. $$ \left|\begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 & 5 \\ 1 & 3 & 6 & 10 & 15 \\ 1 & 4 & 10 & 20 & 35 \\ 1 & 5 & 15 & 35 & 69 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the given 5x5 matrix is \(146\).

Step 1: Prepare the matrix for cofactor expansion

We will perform row operations that involve subtracting a multiple of the first row from other rows so that, except for the first element, all the other elements in the first row become zero. Let's call the matrix A. Perform the following row operations: R2 = R2 - R1 R3 = R3 - R1 R4 = R4 - R1 R5 = R5 - R1 This will result in the following matrix: $$ A = \left|\begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 2 & 5 & 9 & 14 \\ 0 & 3 & 9 & 19 & 34 \\ 0 & 4 & 14 & 34 & 64 \end{array}\right| $$ Step 2: Use cofactor expansion

Step 2: Calculate the determinant using cofactor expansion

Now, expand the determinant along the first row. This can be done by performing the following operation: \(|A| = A_{11}C_{11} + A_{12}C_{12} + A_{13}C_{13} + A_{14}C_{14} + A_{15}C_{15}\) Where \(A_{ij}\) are the elements of the matrix, and \(C_{ij}\) are the corresponding cofactors. But notice that since we have zeros on the first row except for the first element, only the first term in the sum will not be zero. So it simplifies to: $$ |A| = A_{11}C_{11} $$ Now, \(A_{11}\) is 1. To find \(C_{11}\), let's calculate the determinant of the minor that results from removing the first row and first column. $$ C_{11} = \left|\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 2 & 5 & 9 & 14 \\ 3 & 9 & 19 & 34 \\ 4 & 14 & 34 & 64 \end{array}\right| $$ Step 3: Evaluate the 4x4 determinant

Step 3: Calculate the determinant of the 4x4 minor

To evaluate this 4x4 determinant, we can again use cofactor expansion. Let's expand it along the first row, similarly as before: $$ C_{11} = (1) \times \left|\begin{array}{ccc} 5 & 9 & 14 \\ 9 & 19 & 34 \\ 14 & 34 & 64 \end{array}\right| - (2) \times \left|\begin{array}{ccc} 2 & 9 & 14 \\ 3 & 19 & 34 \\ 4 & 34 & 64 \end{array}\right| + (3) \times \left|\begin{array}{ccc} 2 & 5 & 14 \\ 3 & 9 & 34 \\ 4 & 14 & 64 \end{array}\right| - (4) \times \left|\begin{array}{ccc} 2 & 5 & 9 \\ 3 & 9 & 19 \\ 4 & 14 & 34 \end{array}\right| $$ Now we can apply the rule of Sarrus to each 3x3 determinant: $$ C_{11} = (1)(64 - 570 + 476) - (2)(128 - 408 + 180) + (3)(224 - 360 + 84) - (4)(60 - 66) $$ $$ C_{11} = -30 - 100 + 252 + 24 = 146 $$ Step 4: Calculate the final determinant

Step 4: Obtain the final result

Now that we have \(C_{11}\), we can find the determinant of the original 5x5 matrix: $$ |A| = A_{11}C_{11} = (1)(146) = 146 $$ Therefore, the determinant of the given matrix is 146.

Key Concepts

Matrix Row Operations

Matrix row operations are fundamental tools in linear algebra that simplify matrices, making calculations more manageable. By using operations such as row addition, row subtraction, and scalar multiplication, we can transform a matrix into a form that is easier to work with, particularly when computing determinants.

One common goal of these operations is to create zeros in a matrix strategically. For example, when aiming to calculate a determinant using cofactor expansion, it's beneficial to have zeros in the rows or columns. This simplifies the computation significantly.

In our original exercise, we performed row operations by subtracting the first row from the others. This yielded zeros across most of the remaining elements in the first row. These operations do not change the determinant's value but make calculations more straightforward.

Cofactor Expansion

Cofactor expansion, a method used to calculate determinants, is an essential concept in linear algebra. It involves expanding a determinant along a particular row or column. This technique breaks down a matrix into smaller, more manageable parts, known as minors and cofactors.

To perform a cofactor expansion, you choose a row or column to expand. Then, for each element in that row or column, you calculate the determinant of the submatrix that remains after removing that element's row and column. This submatrix is called a minor.

• Each term in the cofactor expansion is the product of the element, its determinant, and a sign factor, which depends on its position in the matrix.

In our exercise, because we had strategically placed zeros in most positions of the first row, the cofactor expansion became much simpler. We only needed to calculate the determinant of one minor, greatly reducing our workload.

4x4 Determinant Calculation

Calculating the determinant of a 4x4 matrix is a step-up in complexity compared to smaller matrices. However, with tools like cofactor expansion and matrix row operations, this task becomes much more manageable.

When dealing with a 4x4 determinant, you often rely on breaking it into smaller 3x3 determinants via cofactor expansion. As demonstrated in our task, after focusing on one row or column to expand upon, you systematically address each minor—itself a 3x3 matrix—and compute its determinant. This hierarchical breakdown helps you avoid overwhelming calculations and errors.

In the given example, the result of the 4x4 determinant was obtained by evaluating several 3x3 determinants and applying the corresponding coefficients and signs. These smaller tasks are executed using the formula for 3x3 determinants or other techniques, such as Sarrus' Rule, until the original 4x4 calculation is complete.

• Remember, each step builds upon the previous ones, culminating in the determinant of the original matrix being computed accurately.
• With practice, breaking down matrices and calculating their determinants using these methods becomes intuitive, empowering you to tackle even larger matrices.