Problem 54 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(A, B, C\) be \(2 \times 2\) matrices with entries from the set of real numbers. Define operation '*' as follows $$ A * B=\frac{1}{2}(A B+B A), \text { then } $$ (A) \(A^{*} I=A\) (B) \(A^{*} A=A^{2}\) (C) \(A * B=B * A\) (D) \(A *(B+C)=A^{*} B+A * C\)

Step-by-Step Solution

Answer

All statements (A), (B), (C), and (D) are true.

Step 1: Understanding the Operation *

The operation * on matrices is defined by \( A * B = \frac{1}{2}(AB + BA) \). This means it is the average of the product of two matrices in both possible orders.

Step 2: Evaluating Statement (A)

For \( A^{*} I = A \), it requires \( \frac{1}{2}(AI + IA) = A \). If \( I \) is the identity matrix, \( AI = A \) and \( IA = A \), so \( \frac{1}{2}(A + A) = A \), which simplifies correctly to \( A \). Hence, statement (A) is true.

Step 3: Evaluating Statement (B)

For \( A^{*} A = A^2 \), it requires \( \frac{1}{2}(AA + AA) = A^2 \), which simplifies to \( A^2 = A^2 \). This is always true, so statement (B) is true.

Step 4: Evaluating Statement (C)

For \( A * B = B * A \), it requires \( \frac{1}{2}(AB + BA) = \frac{1}{2}(BA + AB) \). Since matrix multiplication is not generally commutative, \( AB eq BA \) unless \( A \) and \( B \) commute. Nevertheless, as \( AB + BA = BA + AB \) in terms of addition, this equality always holds. Therefore, statement (C) is true.

Step 5: Evaluating Statement (D)

For \( A * (B+C) = A * B + A * C \), it requires \( \frac{1}{2}(A(B+C) + (B+C)A) = \frac{1}{2}(AB + BA) + \frac{1}{2}(AC + CA) \). Distributing \( A \) and simplifying both sides gives \( \frac{1}{2}(AB + AC + BA + CA) = \frac{1}{2}(AB + BA + AC + CA) \), which is equivalent. Hence, statement (D) is true.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra. It involves multiplying two matrices to produce another matrix. Specifically, when multiplying matrix A by matrix B, each element of the resulting matrix is the sum of the products of the corresponding row elements of A and the column elements of B. To perform matrix multiplication:

• The number of columns in the first matrix (A) must match the number of rows in the second matrix (B).
• If A is an m × n matrix and B is an n × p matrix, the result will be an m × p matrix.
• Each entry in the resulting matrix is calculated as: \(c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj} \).

Understanding matrix multiplication is crucial for engaging with more complex operations and concepts, such as the custom-defined operation "*" in the exercise, which averages the products of matrices. This exercise demonstrates how specific operations, like this averaging, can enhance the application of matrix multiplication properties.

Commutative Property

The commutative property is a fundamental principle in mathematics, meaning the order of the operation does not affect the result. For addition, this property holds universally: \( a + b = b + a \). However, matrix multiplication typically does not adhere to this principle. In other words, for matrices A and B, \( AB eq BA \) in most cases.

However, in this exercise, the custom-defined operation \(A * B = \frac{1}{2}(AB + BA)\) utilizes the commutative nature of matrix addition to redefine matrix multiplication. Although \(AB\) and \(BA\) may not be equal individually, \(AB + BA = BA + AB\) always holds true due to the commutative property of addition. This clever definition makes the new operation \( * \) commutative, which is seen in the statement \(A * B = B * A\). The exercise helps students understand and explore commutative properties in algebra beyond basic operations.

Matrix Addition

Matrix addition is one of the simplest operations that can be performed on matrices. It involves adding corresponding elements from two matrices of the same dimensions. For two matrices A and B, each entry is calculated as:

• \(c_{ij} = a_{ij} + b_{ij}\)

This operation is straightforward and, importantly, commutative. This means that \(A + B = B + A\) for any matrices A and B of the same size.

In the given exercise, the operation defined as \(A * (B+C) = A * B + A * C\) employs matrix addition effectively, demonstrating its compatibility with other matrix operations. Even though the operation "*" entails multiplication, its associated properties rely heavily on the additive nature of matrices to confirm that distribution is maintained. Such exercises can further solidify understanding by showing how matrix addition can integrate seamlessly with multiplication in extended operations.