Problem 33 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(A\) is a non-singular square matrix of order \(n\), then adj \((\operatorname{adj} A)\) is equal to (A) \(|A|^{n} A\) (B) \(|A|^{n-1} A\) (C) \(|A|^{n-2} A\) (D) None of these

Step-by-Step Solution

Answer

The correct answer is (C) \(|A|^{n-2} A\).

Step 1: Understand the Concept of Adjoint

The adjugate (or adjoint) of the matrix \(A\), denoted as \(\operatorname{adj}(A)\), is the transpose of the cofactor matrix of \(A\). For a square matrix of order \(n\), \(\operatorname{adj}(A)\) returns another square matrix of the same order.

Step 2: Apply the Adjoint Theorem for Square Matrices

For any non-singular square matrix \(A\), the identity \(A \cdot \operatorname{adj}(A) = |A| I_n\) holds, where \(I_n\) is the identity matrix of order \(n\) and \(|A|\) is the determinant of \(A\). This identity is crucial to exploring properties of the adjoint.

Step 3: Explore the Relation with Higher Powers

We know \(\operatorname{adj}(\operatorname{adj}(A)) = |A|^{n-2} A\) for a non-singular square matrix \(A\) of order \(n\). This is derived from iterative properties of the determinant with powers in the context of matrices. Ensure that you have seen this property in textbooks or lectures.

Step 4: Choose the Correct Option

From the previous step, we have shown that \(\operatorname{adj}(\operatorname{adj}(A)) = |A|^{n-2} A\). Compare this to the given options and match it with option (C).

Key Concepts

Non-Singular Matrix

When you hear 'non-singular matrix', you're encountering an essential concept in linear algebra. A matrix is non-singular if it has an inverse, and equivalently, a non-zero determinant. Why is this important? Well, such matrices are crucial in solving systems of linear equations, where solutions only exist under specific conditions.

**Key Points:**

• A non-singular matrix is invertible, which means there exists another matrix that, when multiplied with it, gives the identity matrix.
• The determinant of a non-singular matrix is not zero, which ensures that the matrix has full rank.
• Non-singular matrices are significant in theoretical and practical applications, where solving equations or transforming spaces requires invertibility.

Understanding this concept helps distinguish basic properties that define the behavior of matrices in linear transformations.

Cofactor Matrix

The cofactor matrix of a square matrix is at the heart of its adjugate. Let's explore what this means. Each element of the cofactor matrix is obtained by taking the determinant of a submatrix, formed by deleting one row and one column from the original matrix, and then applying a checkerboard pattern of plus and minus signs.

**What You Need to Know:**

• The cofactor of an element is essential in calculating the inverse of a matrix.
• Each cofactor gives insight into how changes in elements affect the matrix's overall properties.
• Beware of the alternating signs pattern applied when constructing the cofactor matrix, as it affects the final values directly.

This matrix not only helps in finding the adjugate by transposing, but also forms the foundation for key algorithms and calculations involving determinants and inverses.

Determinant Properties

Understanding determinant properties is vital for grasping matrix behavior, especially in the context of adjoints and singularity. A determinant is a scalar value that provides insights into the matrix, like invertibility and volume scaling in transformations.

**Crucial Properties:**

• The determinant changes sign when rows or columns are swapped.
• If a matrix has two identical rows or columns, its determinant is zero, indicating singularity.
• For a triangular matrix, whether upper or lower, the determinant is simply the product of its diagonal elements.

These properties are fundamental, especially when dealing with transformations and the relationships between matrices such as adjoints.

Square Matrix

A square matrix is a fundamental structure where the number of rows and columns are equal. This symmetry leads to simplified computations and rich properties in matrix algebra.

**Key Characteristics:**

• Square matrices can be of order \(n \times n\) and they facilitate concepts like determinants and adjugates easily.
• They are a prerequisite for discussing eigenvalues, eigenvectors, and many other advanced concepts.
• These matrices support identities like \(A \cdot \operatorname{adj}(A) = |A| I_n\), which are crucial for solving linear systems.

With their balance of rows and columns, square matrices reveal rich structures and symmetries, playing a central role in both basic and advanced matrix operations.