Problem 21 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If A is a symmetric matrix and \(\mathrm{B}\) is a skew-symmetrix matrix such that \(\mathrm{A}+\mathrm{B}=\left[\begin{array}{cc}2 & 3 \\ 5 & -1\end{array}\right]\), then \(\mathrm{AB}\) is equal to : [April 12, 2019 (I)] (a) \(\left[\begin{array}{cc}-4 & -1 \\ -1 & 4\end{array}\right]\) (b) \(\left[\begin{array}{cc}4 & -2 \\ -1 & -4\end{array}\right]\) (c) \(\left[\begin{array}{ll}4 & -2 \\ 1 & -4\end{array}\right]\) (d) \(\left[\begin{array}{cc}-4 & 2 \\ 1 & 4\end{array}\right]\)

Step-by-Step Solution

Answer

The matrix \( AB \) is \( \begin{bmatrix} 4 & -2 \\ -1 & -4 \end{bmatrix} \), which matches option (b).

Step 1: Identify Properties of Matrices

Recognize that matrix \( A \) being symmetric means \( A^T = A \) and matrix \( B \) being skew-symmetric means \( B^T = -B \). Also, the condition \( A + B = \begin{bmatrix} 2 & 3 \ 5 & -1 \end{bmatrix} \) must be satisfied.

Step 2: Set Up Matrix Equations

Since \( A + B = \begin{bmatrix} 2 & 3 \ 5 & -1 \end{bmatrix} \), assume \( A \) and \( B \) have forms \( A = \begin{bmatrix} a & c \ c & d \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & b \ -b & 0 \end{bmatrix} \), based on their symmetric and skew-symmetric properties.

Step 3: Solve for Symmetric Matrix A

From \( A + B = \begin{bmatrix} 2 & 3 \ 5 & -1 \end{bmatrix} \) and the form of \( B \), write:1. \( a + 0 = 2 \Rightarrow a = 2 \)2. \( c + b = 3 \)3. \( c - b = 5 \)4. \( d + 0 = -1 \Rightarrow d = -1 \).Solve the equations for \( c \) and \( b \):\( c + b = 3 \) and \( c - b = 5 \), so add the equations to get \( 2c = 8 \Rightarrow c = 4 \), subtract to get \( 2b = -2 \Rightarrow b = -1 \).Thus, \( A = \begin{bmatrix} 2 & 4 \ 4 & -1 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \).

Step 4: Calculate AB

Multiply matrices \( A \) and \( B \):\[ AB = \begin{bmatrix} 2 & 4 \ 4 & -1 \end{bmatrix} \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} = \begin{bmatrix} (2 \cdot 0 + 4 \cdot 1) & (2 \cdot -1 + 4 \cdot 0) \ (4 \cdot 0 + -1 \cdot 1) & (4 \cdot -1 + -1 \cdot 0) \end{bmatrix} \].This simplifies to:\[ AB = \begin{bmatrix} 4 & -2 \ -1 & -4 \end{bmatrix} \].

Step 5: Find the Correct Option

The calculated matrix \( AB = \begin{bmatrix} 4 & -2 \ -1 & -4 \end{bmatrix} \) matches option (b).

Key Concepts

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose. This means that when you flip the matrix over its diagonal, you get the same matrix. For example, in a symmetric matrix \( A \), each element satisfies \( a_{ij} = a_{ji} \). This property means that the entries below and above the diagonal are mirror images of each other. Symmetric matrices have some interesting properties, such as all of their eigenvalues being real numbers. Additionally, they play a significant role in various areas of mathematics and physics. Recognizing and understanding symmetric matrices can help simplify complex problems in linear algebra.

Skew-Symmetric Matrix

A skew-symmetric matrix is another type of square matrix with its own unique properties. For a skew-symmetric matrix \( B \), the condition \( B^T = -B \) holds. Essentially, this means if you transpose the matrix and then take the negative, you get the original matrix back. Importantly, all the elements on the main diagonal of a skew-symmetric matrix are zero, that is \( b_{ii} = 0 \). Another key feature is that the eigenvalues of a skew-symmetric matrix come in conjugate pairs, ensuring they are either purely imaginary or zero. Skew-symmetric matrices are useful in various fields, including physics, because they often represent rotational and anti-symmetrical relationships.

Matrix Multiplication

Matrix multiplication involves combining two matrices to form a new matrix. Notably, it involves a dot product between rows of the first matrix and columns of the second. If the first matrix is \( m \times n \) and the second matrix is \( n \times p \), the result will be an \( m \times p \) matrix. Each element \( c_{ij} \) of the resulting matrix is computed as the sum of products: \( c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj} \). It’s crucial to ensure the inner dimensions match for multiplication to be valid. Matrix multiplication is not commutative, which means \( AB eq BA \) generally. However, it plays a key role in transforming spaces, solving linear equations, and many other applications.

Properties of Matrices

Understanding matrix properties is essential because they determine how matrices behave under different operations. For a symmetric matrix \( A \), remember \( A^T = A \) and for a skew-symmetric matrix \( B \), \( B^T = -B \). These properties affect how matrices interact in operations such as addition and multiplication.

• Invariance under Transposition: Symmetric matrices retain their form when transposed.
• Zero Diagonal Elements: Skew-symmetric matrices always have zeros across their main diagonals.
• Real Eigenvalues: Symmetric matrices have real eigenvalues, crucial for physical and engineering applications.
• Multiplicative Non-commutativity: Generally, \( AB eq BA \), indicating order matters in matrix multiplication.

Mastering these properties allows you to approach matrix-related problems with a solid foundation and apply matrix concepts efficiently in various mathematical contexts.