Problem 6 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(A B=A\) and \(B A=B\), where \(A\) and \(B\) are square matrices, then (A) \(B^{2}=B\) and \(A^{2}=A\) (B) \(B^{2}=A\) and \(A^{2}=B\) (C) \(A B=B A\) (D) None of these

Step-by-Step Solution

Answer

The correct answer is (A) \( B^2 = B \) and \( A^2 = A \).

Step 1: Understanding the Equality Conditions

We are given two conditions: \( AB = A \) and \( BA = B \). We need to analyze these equations to derive properties of \( A \) and \( B \).

Step 2: Testing Condition on Matrix A

Multiply both sides of the equation \( AB = A \) by \( A \) from the left and simplify: \( A(AB) = AA \) which simplifies to \( A^2B = A^2 \). Since \( AB = A \), it implies that \( A^2 = A \). This suggests that \( A \) is an idempotent matrix.

Step 3: Testing Condition on Matrix B

Multiply both sides of the equation \( BA = B \) by \( B \) from the right and simplify: \( BAB = BB \) which simplifies to \( BA^2B = B^2 \). Knowing \( ABA = A \) and \( A^2 = A \), we can substitute to find \( B^2 = B \). This suggests that \( B \) is also an idempotent matrix.

Step 4: Concluding the Properties

From Steps 2 and 3, we derived that \( A^2 = A \) and \( B^2 = B \). Checking the options provided, conclusion (A) seems correct as both matrices are idempotent.

Key Concepts

Square Matrices

Square matrices are matrices that have the same number of rows and columns. For a matrix to be considered square, the dimensions must be n x n, where n is any positive integer. These matrices are fundamental in linear algebra and often appear in various applications.

Some key properties make square matrices stand out:

• Determinants: Only square matrices can have determinants, which is a value that can be computed from its elements.
• Identities: The identity matrix, an essential tool in matrix algebra, is always square.
• Invertibility: A square matrix might be invertible (non-singular) or not invertible (singular), depending on its determinant.

Square matrices are vital when discussing operations like multiplication and determinants, making them an integral part of topics like idempotence.

Idempotent Matrices

An idempotent matrix is quite unique because it maintains its form even after being squared. Mathematically, a matrix \( A \) is idempotent if \( A^2 = A \). This means, multiplying an idempotent matrix by itself yields the matrix.

Idempotent matrices have particular characteristics:

• Eigenvalues: The eigenvalues of an idempotent matrix can only be 0 or 1. This is because of the nature of the equation \( A^2 = A \).
• Applications: Idempotency plays a role in areas like statistics, especially in projection matrices used in the linear regression model.
• Density: Unlike general square matrices, the set of idempotent matrices is not dense in the space of all matrices.

Understanding idempotent matrices is crucial for realizing how certain transformations or operations stabilize after repeated applications.

Matrix Multiplication

Matrix multiplication is a cornerstone operation in linear algebra, allowing two matrices to be combined to form a product matrix. Here, the product of two matrices \( A \) and \( B \) is defined only if the number of columns in \( A \) is equal to the number of rows in \( B \). For square matrices, this condition is inherently satisfied.

Matrix multiplication is not as straightforward as the multiplication of numbers and has some unique properties:

• Non-Commutative: Generally, \( AB eq BA \), meaning the order of multiplication matters.
• Associative and Distributive: These property hold for matrix multiplication as \( (AB)C = A(BC) \) and \( A(B + C) = AB + AC \).
• Identity: Multiplying any matrix by its appropriately sized identity matrix (\( I \)) leaves the original matrix unchanged; \( AI = IA = A \).

Learning matrix multiplication is vital for dealing with more complex operations such as computing powers of a matrix, especially in exploring concepts like idempotency.