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

Question

If \(A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]\) and \(A^{2}=\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right]\), 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

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

Step 1: Define Matrix Multiplication for A

Multiply matrix \( A \) by itself: \( A = \begin{bmatrix} a & b \ b & a \end{bmatrix} \). Multiply as follows: the element in row 1, column 1 will be \(a \times a + b \times b\), in row 1, column 2 will be \(a \times b + b \times a\), in row 2, column 1 will be \(b \times a + a \times b\) and in row 2, column 2 will be \(b \times b + a \times a\).

Step 2: Perform Matrix Multiplication

The result of multiplying \( A \) by itself is:\[A^2 = \begin{bmatrix} a^2 + b^2 & 2ab \ 2ab & a^2 + b^2 \end{bmatrix}.\] This matches the structure \( \begin{bmatrix} \alpha & \beta \ \beta & \alpha \end{bmatrix} \) as given in the problem.

Step 3: Compare Resulting Matrix with A^2

By comparing the resulting matrix from Step 2 to \( A^2 = \begin{bmatrix} \alpha & \beta \ \beta & \alpha \end{bmatrix} \), we can identify that \( \alpha = a^2 + b^2 \) and \( \beta = 2ab \).

Key Concepts

Matrix Multiplication

Matrix Multiplication is a way to combine two matrices to produce a new matrix. This operation is fundamental in linear algebra and comes with its own rules. When multiplying two matrices, you take the rows of the first matrix and multiply them by the columns of the second matrix. The resulting values become the elements of a new matrix.
To illustrate, let's consider two matrices, \( A \) and \( B \). The element in the \( i \)-th row and \( j \)-th column of the resulting matrix \( C \) is obtained by multiplying the elements of the \( i \)-th row of \( A \) with the corresponding elements of the \( j \)-th column of \( B \) and then summing up the products. If \( A \) has dimensions \( m \times n \) and \( B \) has dimensions \( n \times p \), the resulting matrix \( C \) will have dimensions \( m \times p \).
Let's apply this to our specific matrices in the original exercise: the matrix for \( A \) is \( \begin{bmatrix} a & b \ b & a \end{bmatrix} \). When multiplying \( A \) by itself, we consider each element's position to compute the new values using this rule. This type of method is crucial for understanding more complex applications such as transformations, solving systems of equations, and more.

Matrix Algebra

Matrix Algebra delves into operations involving matrices and how these operations interact under certain rules. Fundamental operations include addition, subtraction, and multiplication, each playing a key role in various mathematical and real-world applications.
When performing operations on matrices, it is essential to consider the rules regarding dimensions. For example, only matrices of the same dimensions can be added or subtracted. Multiplication has its own set of rules, as previously discussed.

• Addition and Subtraction: This involves adding or subtracting corresponding elements. For example, if two matrices \( A \) and \( B \) are both \( 2 \times 2 \), you simply add or subtract the elements in each corresponding position.
• Scalar Multiplication: This involves multiplying each element of a matrix by a scalar (a single number). It scales the matrix without altering its structure or dimension.
• Matrix Multiplication: As we explored earlier, it involves a more complex calculation, useful for intermediate and advanced algebraic exercises and applications.

Matrix Algebra lays the groundwork for more sophisticated topics, providing tools necessary for exploring vectors, geometry, and transformations.

Symmetric Matrix

A Symmetric Matrix is a specialized type of square matrix that is identical to its transpose. In other words, for a symmetric matrix \( A \), the elements are such that \( a_{ij} = a_{ji} \) for all \( i \) and \( j \). A practical example is when you fold a symmetric matrix along its diagonal from top left to bottom right, both halves would perfectly overlap.
Symmetric matrices have unique properties:

• Real Eigenvalues: A symmetric matrix always has real eigenvalues, which makes it applicable in many fields, particularly in physics and engineering.
• Orthogonal Eigenvectors: The eigenvectors of a symmetric matrix are always orthogonal, which allows spectral decomposition. This is advantageous in solving systems of equations and optimizing computations.
• Matrix Power Property: If a matrix is symmetric, all powers of the matrix are also symmetric, which is why in the given exercise \( A^2 \) remains symmetric.

Symmetric matrices often arise in practical scenarios, such as in the representation of quadratic forms, optimization problems, and statistical measures like covariance matrices. Understanding them can simplify complex problems by reducing them into more manageable components.