Question

Let the matrix \( A = \begin{bmatrix} 1 & 4 \\ 2 & -1 \end{bmatrix} \).
Statement-I: \( A^2 = 9I \)
Statement-II: The eigen values of \( A \) are \( -3 \) and \( 3 \)

• A. Both statement-I and statement-II are true
• B. Both statement-I and statement-II are false
• C. Statement-I is true, but statement-II is false
• D. Statement-I is false, but statement-II is true

Hint:

By the Cayley-Hamilton Theorem, if a matrix satisfies \( A^2 = 9I \), its matrix characteristic equation is \( \lambda^2 - 9 = 0 \). Solving this directly yields eigenvalues \( \lambda = \pm 3 \) instantly without needing separate determinant expansion calculations!

Answer 1

The Correct Option is A

Concept: For any given square matrix \( A \), we can verify matrix relations using direct matrix multiplication. Additionally, the characteristic equation used to find the eigenvalues \( \lambda \) of a matrix \( A \) is determined by solving the determinant equation: \[ \det(A - \lambda I) = 0 \] Alternatively, according to the Cayley-Hamilton theorem, every square matrix satisfies its own characteristic equation. For a \( 2 \times 2 \) matrix, this equation can be simplified as: \[ \lambda^2 - \text{Tr}(A)\lambda + \det(A) = 0 \]

Step 1: Verifying Statement-I by computing \( A^2 \).
We are given the matrix: \[ A = \begin{bmatrix} 1 & 4 \\ 2 & -1 \end{bmatrix} \] Let us compute the square of matrix \( A \), which is defined as the product \( A \times A \): \[ A^2 = \begin{bmatrix} 1 & 4 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} 1 & 4 \\ 2 & -1 \end{bmatrix} \] Performing matrix multiplication row by column explicitly: \[ A^2 = \begin{bmatrix} (1)(1) + (4)(2) & (1)(4) + (4)(-1) \\ (2)(1) + (-1)(2) & (2)(4) + (-1)(-1) \end{bmatrix} \] Evaluating individual components inside the matrix structure: \[ A^2 = \begin{bmatrix} 1 + 8 & 4 - 4 \\ 2 - 2 & 8 + 1 \end{bmatrix} = \begin{bmatrix} 9 & 0 \\ 0 & 9 \end{bmatrix} \] Factoring out the scalar quantity \( 9 \) from the matrix elements: \[ A^2 = 9 \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = 9I \] Hence, \( A^2 = 9I \) is completely accurate, which confirms that

Statement-I is true.

Step 2: Verifying Statement-II by evaluating eigenvalues.
To find the eigenvalues of matrix \( A \), we write down its characteristic equation: \[ \det(A - \lambda I) = 0 \quad \Rightarrow \quad \begin{vmatrix} 1 - \lambda & 4 \\ 2 & -1 - \lambda \end{vmatrix} = 0 \] Expanding the \( 2 \times 2 \) determinant equation: \[ (1 - \lambda)(-1 - \lambda) - (4)(2) = 0 \] Multiplying out the binomial terms carefully: \[ -1 - \lambda + \lambda + \lambda^2 - 8 = 0 \] Combining like terms and constants: \[ \lambda^2 - 9 = 0 \] Solving for \( \lambda \): \[ \lambda^2 = 9 \quad \Rightarrow \quad \lambda = \pm \sqrt{9} \quad \Rightarrow \quad \lambda = 3, -3 \] The calculated eigenvalues of \( A \) are exactly \( 3 \) and \( -3 \). This perfectly matches the text of the second claim. Therefore,

Statement-II is true. Since both individual statements have been analytically verified as correct, Option (A) is the correct choice.