Problem 8 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

EVALUATING DETERMINANTS. $$ \left|\begin{array}{ccc} \sqrt{13}+\sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{26} & 5 & \sqrt{10} \\ 3+\sqrt{65} & \sqrt{15} & 5 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the given 3x3 matrix can be calculated using the diagonal multiplication method as follows: \( (\sqrt{13} + \sqrt{3})(5)(5) + (\sqrt{15})(\sqrt{5})(\sqrt{15} + \sqrt{26}) + (2 \sqrt{5})(\sqrt{10})(3 + \sqrt{65}) \\ - ((\sqrt{5})(5)(3 + \sqrt{65}) + (\sqrt{13} + \sqrt{3})(\sqrt{10})(\sqrt{15}) + (\sqrt{15} + \sqrt{26})(2 \sqrt{5})(5)) \) After evaluating the expression, we will obtain the determinant value.

Step 1: Identify the matrix elements

We have a 3x3 matrix as follows: $$ \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} = \begin{bmatrix} \sqrt{13}+\sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{26} & 5 & \sqrt{10} \\ 3+\sqrt{65} & \sqrt{15} & 5 \end{bmatrix} $$ where \(a_{11} = \sqrt{13}+\sqrt{3}\), \(a_{12} = 2 \sqrt{5}\), \(a_{13} = \sqrt{5}\), \(a_{21} = \sqrt{15}+\sqrt{26}\), \(a_{22} = 5\), \(a_{23} = \sqrt{10}\), \(a_{31} = 3+\sqrt{65}\), \(a_{32} = \sqrt{15}\), and \(a_{33} = 5\).

Step 2: Find the sum of the products of main diagonal and two other diagonals below it

We will now calculate the sum of the products of the main diagonal and two other diagonals below it: \((a_{11}a_{22}a_{33}) + (a_{32}a_{13}a_{21}) + (a_{12}a_{23}a_{31})\) $$ (\sqrt{13} + \sqrt{3})(5)(5) + (\sqrt{15})(\sqrt{5})(\sqrt{15} + \sqrt{26}) + (2 \sqrt{5})(\sqrt{10})(3 + \sqrt{65}) $$

Step 3: Find the sum of the products of the inverse diagonal and two other diagonals above it

Next, we will calculate the sum of the products of the inverse diagonal and two other diagonals above it: \((a_{13}a_{22}a_{31}) + (a_{11}a_{23}a_{32}) + (a_{21}a_{12}a_{33})\) $$ (\sqrt{5})(5)(3 + \sqrt{65}) + (\sqrt{13} + \sqrt{3})(\sqrt{10})(\sqrt{15}) + (\sqrt{15} + \sqrt{26})(2 \sqrt{5})(5) $$

Step 4: Subtract the sum of step 3 from the sum of step 2

Now, we subtract the sum of the inverse diagonal products(products in step 3) from the sum of the main diagonal products(products in step 2) to find the determinant: \( (\sqrt{13} + \sqrt{3})(5)(5) + (\sqrt{15})(\sqrt{5})(\sqrt{15} + \sqrt{26}) + (2 \sqrt{5})(\sqrt{10})(3 + \sqrt{65}) \\ - ((\sqrt{5})(5)(3 + \sqrt{65}) + (\sqrt{13} + \sqrt{3})(\sqrt{10})(\sqrt{15}) + (\sqrt{15} + \sqrt{26})(2 \sqrt{5})(5)) \) After calculating the value of the expression above, we will obtain the determinant of the 3x3 matrix.

Key Concepts

Matrix Algebra

Matrix algebra involves the manipulation and application of matrices, which are rectangular arrays of numbers arranged in rows and columns. These arrays play critical roles in various fields, including engineering, physics, and computer science. Studying matrix operations helps us solve complex problems involving systems of equations, transformations, and more. Key operations in matrix algebra include:

• Addition: Adding corresponding elements from two matrices of the same size.
• Subtraction: Subtracting corresponding elements from one matrix by another.
• Multiplication: Multiplying rows by columns when dimensions are compatible, which is essential in calculating determinants and inverses.
• Finding Determinants: Using specific operations to calculate the determinant, which is pivotal for defining the invertibility of a matrix.
• Inversion: Finding the inverse of a matrix, applied when solving matrix equations and understanding transformations.

Understanding these operations provides the foundation for accessing more advanced concepts, such as eigenvalues and eigenvectors.

Inverse Diagonal

In the context of matrices, especially square matrices, an inverse diagonal refers to the diagonal that stretches from the top right corner to the bottom left corner. It contrasts with the main diagonal, which runs from the top left corner to the bottom right corner.

The inverse diagonal is significant when evaluating determinants, particularly in methods like Sarrus' rule for calculating a 3x3 matrix determinant. This method involves:

• Calculating the product of elements on the inverse diagonal.
• Adding it to products derived from diagonals that go along the inverse diagonal's direction but contain elements from all three rows and columns.
• These calculations contribute to the overall determinant value when paired with similar concepts involving the main diagonal.

By considering both main and inverse diagonals, students can compute determinants more efficiently and grasp the intersections of linear transformations within matrix algebra.

Main Diagonal

The main diagonal of a matrix is composed of elements that stretch from the top left corner to the bottom right corner. This sequence is critical in numerous matrix operations, especially when dealing with square matrices like 3x3 matrices.

In terms of determinant calculation, the main diagonal products play a central role. For any square matrix, the determinant can often be calculated using:

• The product of the elements on the main diagonal.
• Additional diagonals that run parallel and involve a mix of elements from subsequent rows and columns.

For instance, in a 3x3 matrix, the main diagonal helps provide the foundation for the determinant by:

• Serving as a primary element in Sarrus' rule and other determinant calculation techniques.
• Providing direct insight into the presence of zeros or other properties in the matrix that might simplify calculations.

By understanding the significance of the main diagonal, one gains deeper insight into elementary matrix theory and the backbone of matrix transformations.

3x3 Matrix

A 3x3 matrix is a type of square matrix that consists of nine elements arranged in three rows and three columns. These matrices play a vital role in linear algebra, serving as models for systems of equations and linear transformations.

Key characteristics of a 3x3 matrix include:

• 3 Rows and 3 Columns: Defines the structure and size of the matrix.
• Determinant Calculation: Often determined using Sarrus' rule, which involves cross-multiplying the main and inverse diagonals and their corresponding parallel diagonals.
• Invertibility: A 3x3 matrix is invertible (i.e., it has an inverse) if and only if its determinant is non-zero.
• Application in transformations: Used in representing rotations, scaling, and translations within a 2D or 3D space.

Understanding 3x3 matrices provides the foundation for tackling larger matrices and delving into more advanced linear algebra concepts.