Problem 51 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(A\) is a non-singular matrix, then (A) \(A^{-1}\) is symmetric if \(A\) is symmetric (B) \(A^{-1}\) is skew-symmetric if \(A\) is symmetric (C) \(\left|A^{-1}\right|=|A|\) (D) \(\left|A^{-1}\right|=|A|^{-1}\)

Step-by-Step Solution

Answer

The correct options are (A) and (D).

Step 1: Understanding Non-Singular Matrix

A non-singular matrix is one that has an inverse, meaning its determinant is non-zero.

Step 2: Analyzing Symmetries of Inverses

If matrix \( A \) is symmetric, meaning \( A = A^T \), then \( A^{-1} \) is also symmetric due to the property \( (A^{-1})^T = (A^T)^{-1} = A^{-1} \). Therefore, option (A) is true.

Step 3: Evaluating Skew-Symmetries of Inverses

If matrix \( A \) is symmetric, \( A^{-1} \) cannot be skew-symmetric unless it is zero, which contradicts \( A \) being non-singular. Hence, option (B) is false.

Step 4: Examining Determinant Properties

The determinant of the inverse of a matrix \( A \), \( (A^{-1}) \), is \( \left| A^{-1} \right| = \frac{1}{|A|} \). Therefore, option (C) \( \left|A^{-1}\right|=|A| \) is incorrect.

Step 5: Determinant of the Inverse

For a matrix \( A \), \( \left|A^{-1}\right| = \frac{1}{|A|} \) is the correct relation, meaning \( \left|A^{-1}\right|=|A|^{-1} \). Therefore, option (D) is true.

Key Concepts

Non-Singular Matrix

A non-singular matrix, also known as an invertible matrix, is one that has a unique matrix inverse. This property of invertibility is crucial in linear algebra and depends on the determinant of the matrix. A matrix is considered non-singular if its determinant is not zero.

This characteristic of having a non-zero determinant means that the operations involving the matrix are reversible.

• If the determinant of a matrix is zero, it implies that the matrix has no inverse, making it singular.
• For a non-singular matrix, the existence of an inverse, denoted as \(A^{-1}\), is guaranteed. This inverse satisfies the equation \(AA^{-1} = A^{-1}A = I\), where \(I\) is the identity matrix.

Understanding whether a matrix is non-singular helps determine whether solutions to linear equations involving the matrix can be uniquely solved.

Symmetric Matrix

In the realm of matrices, symmetry has a specific meaning. A symmetric matrix is one where the matrix is equal to its transpose. This means for a matrix \(A\), it satisfies \(A = A^T\).

The symmetry of a matrix has a notable implication when it comes to its inverse. If a matrix \(A\) is symmetric, and it is non-singular (meaning it has an inverse), then its inverse \(A^{-1}\) is also symmetric. This arises from the fundamental property of transposes: the transpose of the inverse is the inverse of the transpose, which leads to the equation \((A^{-1})^T = (A^T)^{-1} = A^{-1}\).

• This property is deeply linked with the preservation of symmetry in transformations described by the matrix.
• Symmetric matrices often appear in various fields, such as physics and statistics, where the properties of symmetry can simplify complex problems.

Recognizing symmetric matrices and understanding their properties is vital in many areas of applied mathematics.

Determinant Properties

Determinants play a significant role in matrix theory, providing critical information about the matrix, such as whether it's invertible. For any square matrix \(A\), the determinant is a number that can describe several properties.

One of the crucial properties involves the determinant of a matrix's inverse. For a non-singular matrix \(A\), the determinant of its inverse \(A^{-1}\) is given by \(\left|A^{-1}\right| = \frac{1}{|A|}\). This property denotes that the product of the determinant of a matrix with its inverse is always 1.

• This idea logically expands to: if \( |A| \) is large, \( \left| A^{-1} \right| \) is small, and vice versa.
• The determinant itself is multiplicative, meaning the determinant of the product of two matrices is the product of their determinants.

Understanding these properties is essential in solving linear algebra problems, as determinants help in predicting the behavior and properties of matrix transformations.