Problem 211 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(A\) and \(B\) are symmetric matrices, then show that \(A B\) is symmetric iff \(A B=B A\).

Step-by-Step Solution

Answer

If \(A\) and \(B\) are symmetric matrices, then the product \(AB\) is also symmetric if and only if \(A B = B A\). We have proved both directions of this claim.

Step 1: Understanding the problem

The problem requires the proof of an 'if and only if' statement, thus it consists of two parts: 1. If \(A B = B A\), then \(AB\) is symmetric. 2. If \(AB\) is symmetric, then \(A B = B A\). Each part will be proven individually.

Step 2: Proving the first direction

Assume that \(A B = B A\). The goal is to prove that \(AB\) is symmetric. For this, we need to show \(AB = (AB)^T\). But we know from properties of matrices that \((AB)^T = B^T A^T\). As \(A\) and \(B\) are symmetric matrices, \(B^T = B\) and \(A^T = A\). Therefore, \((AB)^T = BA\). Since we assumed that \(AB = BA\), it implies that \(AB = (AB)^T\), which means that \(AB\) is symmetric.

Step 3: Proving the second direction

Assume that \(AB\) is symmetric. The goal is to prove that \(A B = B A\). If \(AB\) is symmetric, we have \(AB = (AB)^T\), and we know from properties of matrices that \((AB)^T = B^T A^T\). As \(A\) and \(B\) are symmetric matrices, \(B^T = B\) and \(A^T = A\). Therefore, \(BA = AB\). This shows that if \(AB\) is symmetric, then \(A B = B A\).

Step 4: Conclusion

We have thus proved both directions of the 'if and only if' statement, hence completed the proof.

Key Concepts

Matrix Multiplication

Matrix multiplication is a process used in mathematics to multiply two matrices. It involves a series of steps where rows of the first matrix are multiplied with columns of the second matrix. Each element in the resulting matrix is the sum of the products of corresponding elements.

When performing matrix multiplication:

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• If matrix A is of size \(m \times n\) and matrix B is of size \(n \times p\), the resulting matrix, AB, will be of size \(m \times p\).

Matrix multiplication is not commutative, which means \(AB eq BA\) in general. Understanding this helps recognize special cases where commutativity might apply, as in the case of symmetric matrices.

Matrix Transposition

The transpose of a matrix is obtained by swapping its rows and columns. If matrix A is transposed, denoted as \(A^T\), the element in the i-th row and j-th column of A will be in the j-th row and i-th column of \(A^T\).

Important properties of transposed matrices include:

• The transpose of a transpose gets back the original matrix, that is, \((A^T)^T = A\).
• For two matrices A and B, the transpose of their product is the product of their transposes in reverse order: \((AB)^T = B^T A^T\).

Symmetric matrices are a special case where the matrix is equal to its transpose, meaning that \(A^T = A\). This property simplifies some operations, such as verifying the symmetry of matrix products.

Commutative Property

The commutative property in mathematics refers to the ability to change the order of operations and still achieve the same result, usually denoted as \(a\cdot b = b\cdot a\). However, in matrix multiplication, this property does not generally hold. That is, \(AB\) is rarely equal to \(BA\).

However, there are special circumstances under which matrix multiplication is commutative:

• When matrices are symmetric, meaning that they are equal to their transposes.
• In any cases where permutation of the matrices does not change the product's outcome.

In the given exercise, understanding this non-commutative nature is crucial. It forms the base of proving that if \(AB = BA\), then the product \(AB\) is symmetric.

Proof Techniques

Proof techniques are logical steps used to demonstrate the truth of mathematical statements. In the exercise of proving a statement about symmetric matrices, the 'if and only if' logic is used, which involves proving both directions of a statement.

To tackle such proofs:

• "If" part (known as direct proof): Assume one property holds, and deduce the other.
• "Only if" part (often involves indirect proof): Assume the second property holds to show that the first must also be true.

In the solution:- By assuming that \(AB = BA\), the symmetry of the product AB is shown using properties of symmetric matrices.- Conversely, assuming \(AB\) is symmetric shows that \(AB = BA\), thus completing the proof.Applying these logical strategies not only helps in this particular problem but enhances overall problem-solving skills in mathematics.