Problem 26 - Algebra for JEE ADVANCED

Question

If \(A B=A\) and \(B A=B\), then a. \(A^{2} B=A^{2}\) b. \(B^{2} A=B^{2}\) c. \(A B A=A\) d. \(B A B=B\)

Step-by-Step Solution

Answer

All options a, b, c, and d are valid under the given conditions.

Step 1: Understand the Given Conditions

We are given that \(AB = A\) and \(BA = B\). These are equations that describe properties of matrices \(A\) and \(B\) multiplied by each other.

Step 2: Verify Option a. \(A^2 B = A^2\)

We need to verify if \(A^2 B = A^2\) holds true using the given conditions. Consider \(A^2 B = A \cdot A \cdot B\). Using the condition \(AB = A\), substitute \(AB\) with \(A\) in the expression:\[A^2 B = A \cdot (AB) = A \cdot A = A^2\]Hence, option (a) is valid.

Step 3: Verify Option b. \(B^2 A = B^2\)

We need to check if \(B^2 A = B^2\) holds true. Consider \(B^2 A = B \cdot B \cdot A\). Using the condition \(BA = B\), we substitute \(BA\) with \(B\) in the expression:\[B^2 A = B \cdot (BA) = B \cdot B = B^2\]Thus, option (b) is also valid.

Step 4: Verify Option c. \(A B A = A\)

Now, check if \(A B A = A\) is true using the given conditions. Using \(AB = A\), we have:\[A B A = (AB) A = A A = A^2\]Since \(A^2 = A\) (proven in step 2), it follows that \(A B A = A\). Therefore, option (c) is valid.

Step 5: Verify Option d. \(B A B = B\)

Finally, we check if \(B A B = B\). Using \(BA = B\), rewrite the expression:\[B A B = B (AB) = B B = B^2\]Since \(B^2 = B\) (proven in step 3), it follows that \(B A B = B\). Thus, option (d) is valid.

Key Concepts

Matrix multiplication

Matrix multiplication involves the process of taking two matrices, say matrix \( A \) with size \( m \times n \) and matrix \( B \) with size \( n \times p \), and producing a new matrix \( C \) with size \( m \times p \). This operation is well-defined only when the number of columns in the first matrix equals the number of rows in the second matrix. Matrix multiplication is accomplished by taking the dot product of rows from the first matrix with the columns of the second matrix.

Key aspects of matrix multiplication includes:

• Non-commutativity: In general, \( AB eq BA \). Each product can be defined only if dimensions permit, but they are not necessarily identical.
• Associativity: Matrix multiplication is associative, which means \((AB)C = A(BC)\).
• Distributivity: It follows distributive laws such that \( A(B + C) = AB + AC \).

Understanding these properties helps when deciphering complex matrix equations like those seen in matrix properties problems.

Matrix properties

Matrices have unique properties that are fundamental in simplifying and solving matrix equations. Consider the given conditions in the exercise: \( AB = A \) and \( BA = B \). These suggest that matrices \( A \) and \( B \) have idempotent properties, meaning repeating them in operations gives the same result, i.e., \( A^2 = A \) and \( B^2 = B \).

A few important matrix properties are:

• Identity Matrix: An identity matrix, denoted \( I \), acts as a neutral element in multiplication, where \( AI = A \) and \( IA = A \).
• Zero Matrix: Multiplying any matrix with a zero matrix results in a zero matrix.
• Transpose and Symmetric Matrices: The transpose of a matrix results from swapping its rows and columns. A matrix is symmetric if it equals its transpose.

In the context of this problem, recognizing \( A \) and \( B \) as idempotent helps in understanding their role since they satisfy not only \( A^2 = A \) and \( B^2 = B \) but also intertwining relations like \( ABA = A \) and \( BAB = B \).

JEE Advanced mathematics

The Joint Entrance Examination (JEE) Advanced requires a deep understanding of mathematical concepts, often involving complex and abstract topics such as matrix algebra. Matrix algebra problems in JEE tests leverage matrix multiplication, properties, and sometimes interplay those with linear transformations and determinants. The key to excelling in such sections is mastering both foundational rules and advanced theorems of matrices.

For the JEE aspirants:

• Ensure thorough comprehension of fundamental operations, including matrix multiplication and determinants.
• Understand the exceptions and particular identities like the ones involving idempotent and null matrices.
• Practice problem-solving by repeatedly applying matrix properties to diverse and challenging questions.

The exercise explored above demonstrates how basic properties of matrices, such as idempotent nature, can simplify verification of equations and offer shortcuts to solving JEE-level problems. Having a strong grip on such properties can help in tackling questions efficiently and correctly.