Problem 43 - Algebra for JEE ADVANCED

Question

\((-A)^{-1}\) is always equal to (where \(A\) is \(n^{\text {tr-order square matrix })}\) a. \((-1)^{n} A^{-1}\) b. \(-A^{-1}\) c. \((-1)^{n-1} A^{-1}\) d. none of these

Step-by-Step Solution

Answer

The correct answer is b: \(-A^{-1}\).

Step 1: Understanding the Problem

We need to find the inverse of the negative of a matrix, (i.e., if \(-A\), then what is its inverse? This will help us identify the correct option among the given choices.

Step 2: Applying Matrix Properties

The inverse of a product of a scalar and a matrix (e.g., \(-A\)) is given by: \((-A)^{-1} = (-1)^{-1}A^{-1}\). Since ,\((-1)^{-1} = -1\). Therefore, \((-A)^{-1} = -1 \cdot A^{-1} = -A^{-1}\).

Step 3: Comparing Results with Options

Among the given options, we are looking for the expression that matches \(-A^{-1}\). Checking the available options, we find that \(-A^{-1}\) corresponds to option b.

Key Concepts

Square Matrix

A square matrix is a special kind of matrix where the number of rows is equal to the number of columns, denoted as an \(n \times n\) matrix. These matrices are significant because only square matrices have determinants, which is a key factor in many matrix operations, including finding inverses.

• An \(n \times n\) matrix can be represented as \(A = [a_{ij}]\), where both the row index \(i\) and column index \(j\) range from 1 to \(n\).
• Square matrices have distinct properties like being invertible (if they have a non-zero determinant) and having eigenvalues and eigenvectors.
• The identity matrix, a special square matrix with ones on the diagonal and zeros elsewhere, plays a crucial role because multiplying any square matrix by the identity matrix returns the original matrix.

Understanding square matrices is fundamental because they set the stage for more complex operations like matrix inversion, used in solving systems of linear equations and other algebraic functions.

Matrix Scalar Multiplication

Matrix scalar multiplication involves multiplying every element of a matrix by a scalar, which is simply a number. This operation is straightforward and is governed by the distributive property in algebra.

• Given a matrix \(A = [a_{ij}]\) and a scalar \(c\), their product is \(cA = [ca_{ij}]\).
• When the scalar is negative, as in our exercise \(-A\), each element of \(A\) is multiplied by \(-1\), effectively changing the sign of every element.
• This operation does not affect the structure of the matrix; it remains the same size \(n \times m\).

Scalar multiplication is especially useful in transformations and when dealing with inverse relationships, such as multiplying by \(-1\) to handle the negative of a matrix, which is a common task in matrix algebra.

Matrix Properties

Matrix properties are rules and characteristics that matrices follow, valuable for understanding and performing operations like finding inverses. Here are some key properties:

• Additive Identity: Adding the zero matrix to any matrix \(A\) returns \(A\) itself.
• Multiplicative Identity: The identity matrix \(I\), when multiplied by any matrix \(A\), yields \(A\).
• Inverse: The inverse of a matrix \(A\), denoted \(A^{-1}\), satisfies the equation \(AA^{-1} = I\). Only square matrices can have inverses, and not all square matrices are invertible.
• Determinate: The determinant of a matrix helps determine if a matrix is invertible; a non-zero determinant indicates \(A\) has an inverse.

Understanding these properties is essential, as they help navigate through complex algebraic expressions and are the cornerstone for operations such as finding \((-A)^{-1}\) in our problem.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with operations involving matrices. It extends basic arithmetic to higher dimensions and is used extensively in fields like physics, computer science, and statistics.

• Matrix addition and subtraction follow element-wise operations, meaning each element in one matrix is added to or subtracted from the corresponding element in another matrix.
• Matrix multiplication is not element-wise; instead, it involves a row-by-column multiplication method. The product of two matrices is only defined when the number of columns in the first matrix equals the number of rows in the second.
• The concept of inverse and determinant are unique to matrix operations and are crucial for solving linear equations and transforming data.
• In our context, matrix algebra explains why \((-A)^{-1} = -A^{-1}\) is correct, using properties of scalar multiplication and inverse matrices.

Mastery of matrix algebra allows you to solve complex real-world problems and is fundamental for advancing in mathematical theory and applications.