Problem 40 - 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: Understand the Original Determinant

The determinant \( \Delta \) is defined as \( \Delta = \left| \begin{array}{ccc} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{array} \right| \). Each capital letter represents the cofactor of the corresponding small letter. Cofactors are the signed minors of the elements of the matrix.

Step 2: Express the Cofactors

Cofactors for \( a_1, b_1, c_1, \) etc., are calculated from the minor determinants obtained by removing the row and column of each element. This is needed to form the matrix for \( \Delta' \).

Step 3: Construct the Matrix for \( \Delta' \)

The matrix \( \Delta' \) is formed using the cofactors as elements: \( \Delta' = \left| \begin{array}{ccc} A_1 & B_1 & C_1 \ A_2 & B_2 & C_2 \ A_3 & B_3 & C_3 \end{array} \right| \).

Step 4: Apply the Property of Determinants

The matrix consisting of the cofactors of a determinant is known as the adjugate (or adjoint) of the matrix. The property of the determinants states that if the determinant of a matrix is \( \Delta \), the determinant of its adjugate (adjoint) matrix is \( \Delta^{2} \).

Step 5: Calculate the Determinant of the Adjugate

By the property of the adjugate matrix, since \( \Delta' \) is the determinant of the matrix of cofactors, we have \( \Delta' = \Delta^2 \).

Step 6: Choose the Correct Answer

Based on the calculation, the value of \( \Delta' \) is \( \Delta^2 \). Therefore, the correct answer is option (C) \( \Delta^2 \).

Key Concepts

Cofactors

When dealing with determinants, the term "cofactor" often comes up. Cofactors are crucial in understanding how determinants work. Basically, a cofactor is a signed minor of an element in a matrix. To find a cofactor of an element, such as \(a_1\), you need to focus on its position within a matrix.

• First, remove the row and column that the element resides in.
• Then, calculate the determinant of the smaller matrix that remains.
• Finally, apply a sign to the determinant based on the position of the element. This is known as the checkerboard pattern of signs, where the sign of a cofactor is given by \((-1)^{i+j}\), with \(i\) and \(j\) representing the row and column indices of the element.

This process is repeated for each element to construct a new matrix made up of these cofactors. Understanding cofactors helps in finding the adjugate matrix, which plays an essential role in advanced determinant calculations.

Adjugate Matrix

The adjugate matrix, often called the adjoint, is a key concept in linear algebra. It comprises the cofactors of each element in the original matrix, but there's a bit more. Here's how it works in steps:

• Compute the cofactor for each element of the original matrix.
• Place these cofactors into a new matrix, positioning them such that the cofactor of \(a_{ij}\) is placed at the \(ji\) position. This step involves transposing the cofactor matrix.

The adjugate matrix is important because it relates to the inverse of a matrix. Specifically, if you have a matrix \(A\) and its determinant \( \Delta eq 0\), the inverse of \(A\) can be found using the adjugate matrix. In practice, this means multiplying the adjugate matrix by \(1/\Delta\). The property that the determinant of the adjugate is \(\Delta^2\) is utilized in many mathematical applications, including solving systems of linear equations.

Matrix Properties

Matrices possess numerous properties that simplify mathematical computations. A few essential properties relate directly to determinants and adjugates:

• The determinant of a product of two matrices is the product of their determinants. So, if \(A\) and \(B\) are matrices, \(\det(AB) = \det(A)\det(B)\).
• For a matrix \(A\), the determinant of its adjugate (adjoint) matrix is the square of \(\Delta\), the determinant of \(A\). This means \(\det(\text{adj}(A)) = \Delta^2\).
• If a matrix is singular, meaning \(\Delta = 0\), the matrix does not have an inverse.
• The determinant provides crucial information about a matrix, such as its invertibility or whether it represents a transformation that preserves area or volume.

Understanding these properties is fundamental when working with complex mathematical problems involving matrices. They offer insights into matrix behavior and provide ways to compute solutions efficiently.