Problem 28 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Express \(\left|\begin{array}{ccc}1 & 2 & -3 \\ 2 & 1 & 1 \\ 2 & 3 & 1\end{array}\right|^{2}\) in determinant form and find its value also.

Step-by-Step Solution

Answer

The square of the determinant of the given matrix \(A = \left|\begin{array}{ccc}1 & 2 & -3\\ 2 & 1 & 1\\ 2 & 3 & 1\end{array}\right|\) is calculated by first finding the determinant: \(|A| = -14\). Then, we square this value to get the final answer: \(|A|^2 = 196\).

Step 1: Set up the matrix

Write down the given matrix: \[ A = \left|\begin{array}{ccc} 1 & 2 & -3\\ 2 & 1 & 1\\ 2 & 3 & 1 \end{array}\right| \]

Step 2: Calculate the determinant

Use the cofactor expansion method along the first row to compute the determinant of matrix A: \( |A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \) where \(a_{ij}\) are the entries of matrix A, and \(C_{ij}\) are their corresponding cofactors. Determine the cofactors: \( C_{11} = \left|\begin{array}{cc}1 & 1\\3 & 1\end{array}\right| \) \( C_{12} = \left|\begin{array}{cc}2 & 1\\2 & 1\end{array}\right| \) \( C_{13} = \left|\begin{array}{cc}2 & 1\\2 & 3\end{array}\right| \) Calculate the corresponding determinant of each cofactor: \( C_{11} = (1)(1) - (1)(3) = -2 \) \( C_{12} = (2)(1) - (1)(2) = 0 \) \( C_{13} = (2)(3) - (1)(2) = 4 \) Now plug in the values of the entries and their cofactors: \( |A| = (1)(-2) + (2)(0) + (-3)(4) = -2 + 0 - 12 = -14 \)

Step 3: Square the determinant

Now that we have the value of the determinant, we square it to get the final answer: \( |A|^2 = (-14)^2 = 196 \) The square of the determinant of the given matrix is 196.

Key Concepts

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a technique used to calculate the determinant of a matrix. This method involves expanding the determinant along a row or a column of the matrix. For any entry in a matrix, the cofactor can be computed by removing the row and column that contain that entry, forming a smaller matrix.
To execute cofactor expansion, follow these steps:

• Choose a row or column to expand across.
• For each entry in the row or column, compute the minor by omitting the row and column containing that entry.
• Calculate the cofactor by adjusting the sign of the minor according to its position in the matrix.
• Multiply each entry by its corresponding cofactor, and sum these products to obtain the determinant.

In our example, we expanded along the first row of the matrix to find each corresponding cofactor. By doing so, the determinant of the matrix can be computed as shown, providing a pathway to solving for more complex higher-dimensional matrices.

Matrix Algebra

Matrix algebra is a powerful mathematical tool used to handle linear equations and is fundamental in various mathematical computations. A matrix, in essence, is a rectangular array of numbers arranged in rows and columns.
Here are some basic operations within matrix algebra:

• **Addition and Subtraction:** Matrices can be added or subtracted if they have the same dimensions. Each corresponding element is added or subtracted.
• **Multiplication:** Matrix multiplication involves the dot product of rows and columns, requiring that the number of columns in the first matrix matches the number of rows in the second matrix.
• **Determinant:** A scalar value that can be computed from a square matrix which often serves to give information about the matrix, such as invertibility.
• **Transpose:** Flipping a matrix over its diagonal, switching its row and column indices.

Clearing up these matrix operations, especially the calculation of determinants, is crucial in getting familiar with matrix algebra. In our previous steps, determining the determinant involved basic matrix operations and led to calculating the square of the matrix's determinant.

Linear Algebra

Linear Algebra is the field of mathematics that focuses on vectors, vector spaces, linear transformations, and systems of linear equations. It provides a set of tools and techniques that are essential in many fields, such as physics, computer science, and engineering.
Here are some key concepts:

• **Vector Spaces:** Collection of vectors where the operations of addition and scalar multiplication are defined.
• **Linear Transformations:** A function between vector spaces that preserves the operations of vector addition and scalar multiplication.
• **Eigenvalues and Eigenvectors:** Values that provide important insights into the structure of a matrix and are fundamental in matrix decomposition.
• **Systems of Linear Equations:** A collection of linear equations that can be solved simultaneously to find unknown variables.

Understanding Linear Algebra and how matrices operate within it provides the groundwork for solving complex mathematical problems. In our exercise, calculating the determinant and its square utilizes concepts from linear algebra, showing its real-world application in problem-solving.