Problem 2 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(C) \(\Delta^2\)

Step 1: Understanding Cofactors

The elements \(A_1, B_1, C_1, A_2, B_2, C_2, A_3, B_3, C_3\) are the cofactors of the elements \(a_1, b_1, c_1, a_2, b_2, c_2, a_3, b_3, c_3\) respectively in the determinant \(\Delta\). This means that each letter is the result of the determinant left when the row and column are removed from \(\Delta\).

Step 2: Expressing the Determinant in Terms of Cofactors

The determinant \(\Delta^\prime\) has its rows as the cofactors of the original determinant. We know from properties of determinants that if a determinant is formed by the cofactors of another, that determinant is equal to the square of the original determinant, i.e., \(\Delta^\prime = \Delta^2\).

Step 3: Verify by Properties of Determinants

According to Cramer's Rule and properties of determinants, when you take the determinant of the matrix formed by the cofactors of another matrix, the result is the determinant squared: \(\Delta^2\). This confirms our previous result.

Key Concepts

Cofactors

Cofactors are specific values derived from a matrix, which play a crucial role in determining the determinant of a larger matrix. For a given element in a matrix, the cofactor is found by removing the element's row and column, then calculating the determinant of the remaining smaller matrix. This smaller determinant is multiplied by \((-1)^{i+j}\), where \i\ and \j\ are the row and column indices of the element. This multiplication adjusts the sign based on the position within the matrix grid. The use of cofactors is essential in various calculations, including finding the determinant of higher-order matrices and performing matrix inversions. When arranged into a matrix of cofactors, they can reveal deeper characteristics of the original matrix, leading to insights like those found in eigenvalues and eigenvectors. Cofactors also form the basis of the adjugate matrix, an important concept when it comes to calculating the inverse of a matrix, where the transposed matrix of cofactors aids in ensuring the result is accurate.

Cramer's Rule

Cramer's Rule is a method utilized to solve systems of linear equations using determinants. This rule is particularly powerful when the number of equations matches the number of unknowns, as it provides an explicit formula for the solution. The concept is rooted in the idea that a square matrix representing a system of equations can be decomposed into its determinant and individual variable determinants. By expressing each variable in terms of these determinants, Cramer's Rule allows for a straightforward solution method:

• Replace the column of the variable in question with the constants from each equation.
• Calculate the determinant of this modified matrix.
• Divide by the determinant of the original coefficient matrix.

Although elegant, Cramer's Rule can become computationally intensive for larger systems due to the complexity of finding large determinants. Despite this, it serves as a critical theoretical tool in understanding the connections between linear algebra, matrices, and systems of equations.

Properties of Determinants

Determinants possess several properties that are fundamental to their real-world applications. Understanding these properties can simplify a wide range of mathematical problems, including matrix transformations, finding volumes in geometry, and solving systems of linear equations. Key properties of determinants include:

• Interchanging two rows (or columns) negates the determinant.
• Multiplying a row (or column) by a scalar multiplies the determinant by the same scalar.
• If two rows or columns of a matrix are identical, the determinant is zero.
• The determinant of a product of matrices equals the product of their determinants.

One particularly useful property is that the determinant of a matrix constructed from the cofactors of another matrix is equal to the square of the original determinant. This property underpins many advanced applications in linear algebra, including the verification of matrix inverses and matrix equation solutions. Indeed, understanding and leveraging these properties is vital in both theory and practical computations.