Problem 64 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]\) and \(A^{2}=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha\end{array}\right]\), then (A) \(\alpha=a^{2}+b^{2}, \beta=a b\) (B) \(\alpha=a^{2}+b^{2}, \beta=2 a b\) (C) \(\alpha=a^{2}+b^{2}, \beta=a^{2}-b^{2}\) (D) \(\alpha=2 a b, \beta=a^{2}+b^{2}\)

Step-by-Step Solution

Answer

Option (B): \(\alpha = a^2 + b^2, \beta = 2ab\).

Step 1: Understanding Matrix Squaring

We need to square matrix \( A \) given by \( A = \begin{bmatrix} a & b \ b & a \end{bmatrix} \). Squaring a matrix involves multiplying it by itself: \( A^2 = A \times A \).

Step 2: Multiplying the Matrices

Multiply matrix \( A \) by itself:\[A^2 = \begin{bmatrix} a & b \ b & a \end{bmatrix} \times \begin{bmatrix} a & b \ b & a \end{bmatrix}.\]Calculating each element:- Top-left element: \( a^2 + b^2 \)- Top-right element: \( ab + ba = 2ab \)- Bottom-left element: \( ba + ab = 2ab \)- Bottom-right element: \( b^2 + a^2 \).

Step 3: Writing the Squared Matrix

The resulting matrix \( A^2 \) is:\[A^2 = \begin{bmatrix} a^2 + b^2 & 2ab \ 2ab & a^2 + b^2 \end{bmatrix}.\]This matches the format \( \begin{bmatrix} \alpha & \beta \ \beta & \alpha \end{bmatrix} \), with \( \alpha = a^2 + b^2 \) and \( \beta = 2ab \).

Step 4: Comparing with Given Options

Comparing with the given options, we find that option (B) where \( \alpha = a^2 + b^2 \) and \( \beta = 2ab \) matches our calculations.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, involving the product of two matrices. When multiplying matrices, it is crucial to ensure the number of columns in the first matrix matches the number of rows in the second one. Otherwise, the multiplication is undefined. The process entails multiplying each element of the rows of the first matrix with the corresponding elements of the columns of the second matrix, then summing these products to form a single element in the resulting matrix. This operation is not commutative, meaning the order of the matrices matters.

• Matrix multiplication is associative: \( (AB)C = A(BC) \).
• It is distributive: \( A(B+C) = AB + AC \).
• Generally, \( AB eq BA \), except in special circumstances like identity matrices.

Matrix multiplication forms the backbone of many operations in linear algebra, including transformations and systems of equations.

Matrix Squaring

Matrix squaring is a specific example of matrix multiplication where a matrix is multiplied by itself. This operation is symbolized as \( A^2 \). The condition for squaring is, naturally, that the matrix must be square (the number of rows equals the number of columns).Consider a matrix \( A \) such that:\[A = \begin{bmatrix} a & b \ b & a \end{bmatrix}\]Squaring \( A \) involves multiplying it by itself:\[A^2 = A \times A\]Each element of the resulting matrix is found by following the typical matrix multiplication steps.

• Top-left and bottom-right elements: \( a^2 + b^2 \)
• Top-right and bottom-left elements: \( 2ab \)

This results in a symmetric matrix, demonstrating that the operation of squaring can sometimes preserve or create symmetry.

Symmetric Matrix

A symmetric matrix is one where the matrix is equal to its transpose. This means that the elements on or around the diagonal are mirrored, making \( A = A^T \). The main diagonal remains constant, while the elements reflect across this line.In the context of our problem, the matrix \( A \) has been squared to produce another symmetric matrix:\[A^2 = \begin{bmatrix} a^2 + b^2 & 2ab \2ab & a^2 + b^2 \end{bmatrix}\]Here, both the original matrix \( A \) and the resulting matrix \( A^2 \) were symmetric. This holds true for any square matrix where the elements outside the main diagonal are mirrors of each other.

• Symmetric matrices arise naturally in many applications, such as energy minimization problems and covariance matrices.
• They simplify calculations, especially in eigenvalue problems.

Linear Algebra

Linear Algebra is the branch of mathematics concerning vector spaces, and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but it also delves deeper into understanding matrices and their transformations. Matrices, a core component of linear algebra, help encode information about linear transformations. They are compact and offer a convenient representation for multi-dimensional operations, such as those encountered when solving systems of equations or performing complex transformations. Key concepts in linear algebra include:

• Vectors and vector spaces: Collections of vectors follow linear operations within defined spaces.
• Functions and mappings: Linear transformations that can be expressed as matrices.
• Determinants and inverses: Tools to evaluate and manipulate matrices in solving equations and understanding transformations.

Understanding linear algebra gives you the tools to engage with complex problem-solving scenarios, from physics and engineering to computer science and statistics.