Problem 32 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(A\) is an invertible matrix, then (A) \(\operatorname{adj} A^{\prime}=(\operatorname{adj} A)^{\prime}\) (B) \(\operatorname{adj} A^{\prime}=\operatorname{adj} A\) (C) adj \(A^{\prime}=A^{\prime}\) (D) None of these

Step-by-Step Solution

Answer

Option (A) is correct: \(\operatorname{adj} A' = (\operatorname{adj} A)'\).

Step 1: Understanding the Problem

We need to determine the correct statement regarding the adjugate (adjoint) of the transpose of an invertible matrix \(A\). We have four options to choose from.

Step 2: Recall Definitions

The transpose of a matrix \(A\), denoted \(A'\), is the matrix obtained by swapping the rows and columns. The adjugate matrix of \(A\), \(\operatorname{adj} A\), is the transpose of the cofactor matrix of \(A\).

Step 3: Using Properties of Adjugate

For any matrix \(A\), \(\operatorname{adj}(A') = (\operatorname{adj} A)'\) holds true. This states that the adjugate of a transpose matrix \(A'\) is equal to the transpose of the adjugate of matrix \(A\).

Step 4: Evaluate Each Option

- Option (A): \(\operatorname{adj} A' = (\operatorname{adj} A)'\) is true, as per the property stated above. - Option (B): \(\operatorname{adj} A' = \operatorname{adj} A\) is not generally true. - Option (C): \(\operatorname{adj} A' = A'\) is incorrect because adjugates are not equal to the matrix transpose itself unless \(A\) is an identity matrix. - Option (D) is incorrect since option (A) is correct.

Step 5: Conclusion

Option (A) is the correct choice, based on the given properties of adjugate matrices for an invertible matrix \(A\).

Key Concepts

Invertible Matrix

An invertible matrix, often called a non-singular or non-degenerate matrix, is one that can be "reversed" or "undone" by another matrix. If a square matrix is invertible, it means there exists another matrix which, when multiplied with the original matrix, results in the identity matrix. Mathematically, if we have a matrix \( A \), the matrix is invertible if there exists a matrix \( B \) such that:

• \( AB = BA = I \)

Here, \( I \) is the identity matrix, which acts like the number 1 in matrix operations. A key property of invertible matrices is that they have a non-zero determinant. If the determinant of a matrix is zero, it implies the matrix is not invertible. In real-world applications, invertible matrices are used in solving systems of linear equations, among other things.

Adjugate Matrix

The adjugate matrix, also referred to as the adjoint matrix, plays an important role in finding the inverse of a matrix. For a given square matrix \( A \), the adjugate is the transpose of the cofactor matrix of \( A \). It is often denoted as \( \operatorname{adj} A \). The adjugate matrix is useful because, for any invertible matrix \( A \), the inverse can be expressed using the adjugate matrix and the determinant of \( A \):

• \( A^{-1} = \frac{1}{\det(A)} \operatorname{adj}(A) \)

This relationship shows how the adjugate is directly used to compute the inverse, which is essential in operations that require reversing transformations like systems of equations, graphics, and more. Remember, though, that the adjugate matrix on its own does not simplify the operations unless paired with the determinant.

Matrix Transpose

The transpose of a matrix is a fundamental concept in linear algebra. Transposing a matrix means flipping it over its diagonal. This operation turns the rows of a matrix into columns and vice versa. For a matrix \( A \), the transpose is denoted as \( A' \) or \( A^T \). If \( A \) is a matrix of size \( m \times n \), then \( A' \) will be of size \( n \times m \).

• The main diagonal remains unchanged during transposition.
• Transposition is a simple transformation used in various matrix operations.

A special property of transpose matrices is \( (A')' = A \), meaning if you transpose a matrix twice, you return to the original. The transpose operation is frequently used in mathematics and engineering, especially where matrices need to be transformed or manipulated for various calculations.

Cofactor Matrix

The cofactor matrix is an essential step in determining both the determinant and the adjugate of a matrix. For a given square matrix \( A \), the cofactor matrix is formed by replacing each element with its corresponding cofactor. A cofactor is a signed minor, computed by taking the determinant of a submatrix obtained by removing the associated row and column.

• Each cofactor is calculated as \( (-1)^{i+j} \det(M_{ij}) \), where \( M_{ij} \) is the submatrix.

The arrangement of these cofactors creates the cofactor matrix, often denoted as \( C \).
The process of finding minors and determining signs (positive or negative) is determined by the position of the element. After obtaining the cofactor matrix, the transpose of this matrix forms the adjugate, a crucial component when computing the inverse of a matrix. This whole process illustrates the complex interrelationships within matrix operations that allow for solving diverse mathematical tasks.