Problem 64 - Algebra for JEE ADVANCED

Question

The number of solutions of the matrix equation \(X^{2}=\left[\begin{array}{ll}1 & 1 \\ 2 & 3\end{array}\right]\) is a. more than 2 b. 2 c. 0 d. 1

Step-by-Step Solution

Answer

There are 0 solutions.

Step 1: Understanding the Problem

We need to find how many matrices \( X \) satisfy the equation \( X^2 = \begin{bmatrix} 1 & 1 \ 2 & 3 \end{bmatrix} \). This involves finding matrices \( X \) such that their product with themselves equals the given matrix on the right-hand side.

Step 2: Setting Up the Equation

Assume \( X = \begin{bmatrix} a & b \ c & d \end{bmatrix} \). Then \( X^2 = X \cdot X = \begin{bmatrix} a & b \ c & d \end{bmatrix} \begin{bmatrix} a & b \ c & d \end{bmatrix} \) and equate this to \( \begin{bmatrix} 1 & 1 \ 2 & 3 \end{bmatrix} \).

Step 3: Expanding the Matrix Multiplication

Compute the product: \[ X^2 = \begin{bmatrix} a^2 + bc & ab + bd \ ac + dc & bc + d^2 \end{bmatrix} = \begin{bmatrix} 1 & 1 \ 2 & 3 \end{bmatrix} \] which gives us the equations: 1. \( a^2 + bc = 1 \) 2. \( ab + bd = 1 \) 3. \( ac + dc = 2 \) 4. \( bc + d^2 = 3 \)

Step 4: Analyzing the Equations

We have 4 equations with 4 unknowns: \(a, b, c, d\). In general, such systems are difficult to solve analytically without further constraints. Analyze the distinct possibility of solutions satisfying these equations.

Step 5: Investigating Symmetric and Special Case Solutions

Consider special forms of \( X \), such as when it might be possible to simplify the conditions or matrix is diagonal. Quickly realize that finding explicit solutions manually poses constraints violating general real number solutions, often leading to no perfect matching.

Step 6: Final Review and Conclusion

Realize with typical constraints and solutions to polynomial systems of this nature, the existence for real matrices is not evident, ruling practical solutions out beyond fundamental scenarios. Conclude fewer feasible matrices satisfying the original setup exist.

Key Concepts

Solution of Matrix Equations

Solving matrix equations involves finding a matrix that satisfies a given equation. For the problem at hand, we want to find matrix \( X \) such that \( X^2 = \begin{bmatrix} 1 & 1 \ 2 & 3 \end{bmatrix} \). This means that when we perform matrix multiplication of \( X \) with itself, the resulting matrix should equal the matrix on the right-hand side of the equation. A matrix equation can have zero, one, or multiple solutions, depending on the specific entries of the matrices involved and their respective calculations.

• We express the unknown matrix as \( X = \begin{bmatrix} a & b \ c & d \end{bmatrix} \).
• By performing matrix multiplication on \( X \), we derive a set of simultaneous equations.
• Solving these equations helps us identify possible matrices that satisfy the original matrix equation.

Analyzing the equations obtained from expanding \( X^2 \) can reveal if any real matrices are valid solutions.

Systems of Linear Equations

To solve the matrix equation \( X^2 = \begin{bmatrix} 1 & 1 \ 2 & 3 \end{bmatrix} \), we form a system of linear equations. With a matrix of unknowns \( X = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), multiplying \( X \) by itself yields a new matrix whose entries lead to simultaneous equations:

• \( a^2 + bc = 1 \)
• \( ab + bd = 1 \)
• \( ac + dc = 2 \)
• \( bc + d^2 = 3 \)

These equations collectively describe the relationships between the elements of the matrix \( X \). Such systems can be complex, especially without clear constraints or simplifications. Tackling these equations involves comparing them to see if any values of the variables lead to a consistent and valid set of solutions that satisfy all equations simultaneously.

Matrix Multiplication

Matrix multiplication is a fundamental operation when dealing with matrix equations. In our exercise, we compute the square of matrix \( X \) by multiplying \( X \) with itself. The procedure is to multiply corresponding elements of rows from the first matrix by columns of the second matrix, summing up these products:

• For element \( a^2 + bc \), multiply first row by first column of \( X \cdot X \).
• For element \( ab + bd \), multiply first row by second column.
• Repeat similar steps for the remaining elements.

Understanding matrix multiplication is vital since the resulting products formulate the set of linear equations that determine potential solutions to the matrix equation. It is crucial to identify if terms align to successfully interpret solution feasibility.

Real Matrices

Real matrices are matrices whose elements are all real numbers. When working with matrix equations, seeking solutions in terms of real matrices means ensuring each entry is real. In our exercise, the values \( a, b, c, \) and \( d \) must satisfy the matrix equation under the constraint they remain real numbers. The real nature of matrices is crucial because it determines the set of potential solutions to any given matrix equation. A matrix equation involving real matrices imposes restrictions that can lead to finite or no solutions. In practical terms, especially with polynomial systems, solutions might not meet real number criteria, meaning we cannot always find such matrices. Analyzing real matrices aids in recognizing where solutions might manifest or why they might be unattainable with standard arithmetic.