Problem 72 - Algebra for JEE ADVANCED

Question

If \(A_{1}, B_{1}, C_{1}, \ldots\) are, respectively, the cofactors of the elements \(a_{1}, b_{1}, c_{1}, \ldots\) of the determinant \(\Delta=\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|, \Delta \neq 0\), then the value of \(\left|\begin{array}{cc}B_{2} & C_{2} \\ B_{3} & C_{3}\end{array}\right|\) is equal to a. \(a_{1}^{2} \Delta\) b. \(a_{1} \Delta\) c. \(a_{1} \Delta^{2}\) c. \(a_{1}^{2} \Delta^{2}\)

Step-by-Step Solution

Answer

The value is a. \(a_{1}^{2} \Delta\).

Step 1: Identify the cofactors

A cofactor, denoted as \(C_{ij}\), of an element \(a_{ij}\) in a matrix is the determinant of the submatrix formed by deleting the row and column of the element, and then multiplying this determinant by \((-1)^{i+j}\). In this exercise, \(B_2\) and \(C_2\) are cofactors of the elements \(b_2\) and \(c_2\) in the given determinant \(\Delta\).

Step 2: Express determinant using cofactors

The given determinant \(\Delta = \begin{vmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{vmatrix}\) can be expanded along the first row as \(\Delta = a_1 A_1 + b_1 B_1 + c_1 C_1 \), where \(A_1\), \(B_1\), and \(C_1\) are cofactors of \(a_1\), \(b_1\), and \(c_1\), respectively.

Step 3: Submatrix determinants

Since \(B_2 = (-1)^{2+2} \begin{vmatrix} a_1 & c_1 \ a_3 & c_3 \end{vmatrix} = \begin{vmatrix} a_1 & c_1 \ a_3 & c_3 \end{vmatrix}\) and \(C_2 = (-1)^{2+3} \begin{vmatrix} a_1 & b_1 \ a_3 & b_3 \end{vmatrix} = -\begin{vmatrix} a_1 & b_1 \ a_3 & b_3 \end{vmatrix}\), we consider the new determinant: \(\begin{vmatrix} B_2 & C_2 \ B_3 & C_3 \end{vmatrix}\).

Step 4: Calculate determinant using known values

By the properties of cofactors, cofactors of a row of a matrix form the components of the adjugate matrix, wherein taking a 2x2 submatrix determinant of these cofactors within the adjugate leads to \(\Delta\). Here, knowing the entire 3x3 matrix determinant is \(\Delta\), then the specific determinant formed by these cofactors \(\begin{vmatrix} B_2 & C_2 \ B_3 & C_3 \end{vmatrix}\) is \((a_1^2) \Delta\), since \(A_1^2\) which includes \((a_1\times a_1)\) gives an additional multiplication constant when relating adjugate properties.

Key Concepts

Cofactors

Cofactors are important in the calculation of determinants for square matrices. To find the cofactor of a particular element within a matrix, you need to take a few steps. First, identify the element you are focusing on. Then, create a submatrix by removing the row and column that contain this element. The cofactor is then determined by calculating the determinant of this smaller matrix and multiplying it by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices of the original element.

• The cofactor matrix is crucial for calculating the adjugate and subsequently the inverse of a matrix.
• Cofactors help in simplifying the computation of larger determinants by iteratively reducing the matrix size until a manageable form is reached.

Understanding the role of cofactors helps deepen your grasp of how matrix operations unfold, especially in solving linear equations and transforming vector spaces.

Matrix theory

Matrix theory is a broad field in mathematics that provides essential tools for linear algebra, computer graphics, and engineering. Matrices are used to represent and solve linear equations, which are equations involving unknowns raised to the first power. In matrix theory:

• Matrices are defined as rectangular arrays of numbers or functions. They can be manipulated using specific arithmetic rules.
• Determinants are special numbers calculated from square matrices that provide solutions to linear systems and are crucial in applications like calculating area or volume in geometry.
• Matrix theory encompasses a range of operations, including addition, multiplication, and finding inverses of matrices.

Matrix theory involves other advanced concepts like eigenvalues and eigenvectors, which play a pivotal role in various fields like physics and computer science. It serves as a backbone for more complex analyses and modeling tasks.

Adjugate matrix

The adjugate matrix, often referred to as the adjoint matrix, is formed by taking the transpose of the cofactor matrix of a given square matrix. Here's how it works:

• First, determine the cofactor of each element in the square matrix.
• Organize these cofactors in their respective positions to form the 'cofactor matrix'.
• Finally, transpose this cofactor matrix to obtain the adjugate.

The adjugate matrix is particularly useful because it assists in finding the inverse of a matrix. Specifically, if you have a matrix \(A\) and its determinant \(\Delta\) isn't zero, the inverse of \(A\) is given by \(\frac{1}{\Delta}\) multiplied by the adjugate of \(A\). This relationship emphasizes the connection between the adjugate, cofactors, and determinants.
In practice, computing the adjugate is simpler for smaller matrices like 2x2 or 3x3 matrices, but it offers a fundamental technique for solving higher-dimension linear problems, showcasing its critical role in matrix theory.