Problem 2 - A Complete Resource Book in Mathematics for JEE Main

Question

If the capital letters denote the cofactors of the corresponding small letters in the determinant \(\Delta=\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|\) then the value of \(\Delta^{\prime}=\left|\begin{array}{ccc}A_{1} & B_{1} & C_{1} \\\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|\) is (A) 0 (B) \(2 \Delta\) (C) \(\Delta^{2}\) (D) \(\Delta\)

Step-by-Step Solution

Answer

The value of \(\Delta'\) is \(\Delta^2\) (option C).

Step 1: Understanding Determinant and Cofactors

Given the determinant \(\Delta\) with cofactor matrix notation: each element in the cofactor matrix is the cofactor of the corresponding element in the original matrix.

Step 2: Cofactor Matrix is Transpose of Adjunct

The cofactor matrix \(\begin{bmatrix} A_1 & B_1 & C_1 \ A_2 & B_2 & C_2 \ A_3 & B_3 & C_3 \end{bmatrix}\) is actually the transpose of the adjugate (adjoint) of the matrix \(\begin{bmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{bmatrix}\).

Step 3: Formula for Determinant of Adjunct

The determinant of the adjugate of a matrix is the determinant of the matrix itself raised to the power of \(n-1\), where \(n\) is the order of the matrix. Given that the determinant \(\Delta\) is a 3x3 matrix, the order \(n=3\).

Step 4: Calculating \(\Delta'\)

For a 3x3 matrix, the determinant of the cofactor matrix \(\Delta'\) is \(\Delta^2\) because the order is 3. Therefore, \(\Delta' = \Delta^2\).

Key Concepts

Cofactor Matrix

The cofactor matrix is a critical concept in matrix algebra. It's formed by replacing each element of a matrix with its corresponding cofactor. A cofactor, in turn, is a minor of the matrix element, with a sign based on its position. In a 3x3 matrix, each cofactor is calculated by deleting the row and column of the element and finding the determinant of the resulting 2x2 matrix.

• The sign is determined by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices of the element.
• This means if the element is in an even position, its cofactor retains the sign of the minor's determinant, while an odd position implies a sign change.

Understanding the cofactor matrix is essential because it is used to ultimately find the adjugate of a matrix, which is pivotal in various matrix operations.

Adjugate

The adjugate (or adjoint) of a matrix is closely related to the cofactor matrix. The adjugate is essentially the transpose of the cofactor matrix. Transposing a matrix involves swapping its rows with columns, effectively rotating it.

• For the adjugate of a 3x3 matrix, once you compute all the cofactors and form the cofactor matrix, you transpose this matrix.
• This process is crucial for calculating the inverse of a matrix using the formula \(A^{-1} = \frac{1}{\Delta} \text{adj}(A)\), where \(\Delta\) is the determinant of the original matrix.

This relationship between the cofactor matrix and the adjugate highlights the interconnectedness of these matrix concepts and their importance in solving matrix equations.

3x3 Matrix Determinant

Calculating the determinant of a 3x3 matrix involves a specific formula that utilizes the diagonal elements and their cofactors. For a 3x3 matrix \(\begin{bmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{bmatrix}\), the determinant \(\Delta\) is calculated as follows:

• \(\Delta = a_1(b_2c_3 - b_3c_2) - b_1(a_2c_3 - a_3c_2) + c_1(a_2b_3 - a_3b_2)\).
• This formula requires finding minors and applying them strategically to find the overall determinant.

The determinant is a key characteristic of the matrix, which among other uses, helps determine if a matrix is invertible (nonzero determinant) or singular (zero determinant). Its value is also vital in scalar transformations of the adjugate matrix.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with the study and manipulation of matrices. It's a foundational tool in various fields such as computer science, physics, and engineering. Some key operations that define matrix algebra include:

• Matrix Addition: Matrices can be added if they have the same dimensions, by adding corresponding elements.
• Matrix Multiplication: More complex than addition, it involves multiplying rows by columns and is feasible when the number of columns in the first matrix equals the number of rows in the second.
• Determinants: These give insight into numerous properties of the matrix, including its invertibility.
• Inverses: Matrices are inverted to solve systems of equations and are denoted commonly as \(A^{-1}\).

Understanding these basic operations and concepts such as cofactors, adjugates, and the determinant is essential for mastering matrix algebra.