Problem 8 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(\Delta=\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|\) and \(A_{1}, B_{1}, C_{1}\) are cofactors of \(a_{1}, b_{1}, c_{1}\), respectively, then determinant \(\left|\begin{array}{lll}A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|=\) (a) \(\Delta\) (b) \(\Delta^{2}\) (c) \(\Delta^{3}\) (d) 0

Step-by-Step Solution

Answer

The determinant is \( \Delta^2 \), option (b).

Step 1: Understand the Problem

We need to find the value of the determinant of a matrix formed by the cofactors of the elements of the first row of a given matrix. The original matrix has determinant \( \Delta \).

Step 2: Recall Properties of Determinants

One important property of determinants is that if a matrix \( A \) is \( n \times n \) and \( n > 1 \), its determinant \( \Delta \) has properties related to its cofactors. The cofactor matrix of \( A \), also known as the adjugate or adjoint of \( A \), when multiplied with \( A \), yields \( \text{adj}(A) \cdot A = \Delta I \), where \( I \) is the identity matrix.

Step 3: Apply the Determinant of Cofactor Matrix

The determinant of the cofactor matrix (or adjugate matrix) of a matrix is equal to the determinant of the original matrix raised to the power of \( n-1 \), where \( n \) is the order of the matrix. In this case, \( n = 3 \).

Step 4: Calculate the Determinant for 3x3 Matrix

Given that \( n = 3 \), we use the property \( \left| \text{adj}(A) \right| = \Delta^{n-1} \) which implies \( \left| \text{adj}(A) \right| = \Delta^{3-1} = \Delta^2 \).

Step 5: Correct Answer

Therefore, the determinant \( \left| \begin{array}{lll} A_{1} & B_{1} & C_{1} \ A_{2} & B_{2} & C_{2} \ A_{3} & B_{3} & C_{3} \end{array} \right| \) is \( \Delta^2 \), which corresponds to option (b).

Key Concepts

Cofactors

In linear algebra, a "cofactor" is a crucial element when dealing with matrices, especially when calculating determinants. A cofactor is associated with a specific element in a matrix and is involved in finding the determinant of that matrix. To find the cofactor of an element, first, remove the row and column containing the element. Then, compute the determinant of the resulting smaller matrix. Finally, apply a sign based on the position of the element: positive if the sum of its row and column indices is even, and negative if odd. This can be expressed using the formula:

• For an element located at the \(i,j\) position, the cofactor \(C_{ij}\) is \( (-1)^{i+j} \, times, \, the \, determinant \, of \, the \, minor\).

Cofactors are the building blocks for determining the larger determinant. They serve as multipliers when expanding the determinant across a row or a column. Understanding cofactors is crucial as they form the elements of something called the "adjugate matrix," which plays a vital role in matrix inversions and solving systems of linear equations.

Adjugate Matrix

The adjugate matrix, sometimes called the adjoint matrix, is formed by the cofactors of a given square matrix. To construct the adjugate matrix, one calculates the cofactor for each element of the matrix and arranges them in a transposed manner. If matrix \(A\) is an \(n \, x \, n\) matrix, its adjugate \(\text{adj}(A)\) has the following properties:

• Each element of \(\text{adj}(A)\) is the cofactor of the transposed position from matrix \(A\).
• The relationship between the original matrix \(A\) and its adjugate is given by \(\text{adj}(A) \cdot A = \Delta I\), where \(\Delta\) is the determinant of \(A\) and \(I\) is the identity matrix.

The adjugate matrix is particularly important because it provides a mechanism to compute the inverse of a matrix, assuming the matrix is not singular (i.e., its determinant is not zero). The inverse can be expressed as \(A^{-1} = \frac{\text{adj}(A)}{\Delta}\). Thus, the adjugate matrix serves as a cornerstone in many advanced matrix operations, underpinning solutions to linear equations and analysis of mathematical models.

Properties of Determinants

Determinants are a fundamental concept in linear algebra, as they provide key insights into a matrix. These properties can simplify calculations and help us understand the nature of the matrix:

• Linearity: The determinant function is linear in each row and each column separately. This means you can factor out constants and split sums when dealing with determinants.
• Row Switching: Swapping two rows or two columns of a matrix flips the sign of its determinant. This property is instrumental when performing row operations to simplify determinants.
• Zero Rows or Columns: If a matrix has a zero row or a zero column, its determinant immediately equals zero. This shortcut can save substantial time.
• Determinant of the Identity: The determinant of an identity matrix is always 1, regardless of its size.
• Multiplicative Property: The determinant of the product of two matrices equals the product of their determinants, \(\left|AB\right| = \left|A\right| \cdot \left|B\right|\).

Understanding these properties not only aids in manual computations of determinants but also enhances the grasp of matrix behavior during various algebraic operations. In sum, determinants act as a tool to assess potential changes in systems and to solve complex systems of equations efficiently.