Question

If \( A = \begin{pmatrix} -3 & 2 \\ 1 & 0 \end{pmatrix} \) is a \( 2 \times 2 \) matrix, then \( A \) satisfies the relation ...........

• A. \( A^2 - 2A + 3I = 0 \)
• B. \( A^3 - A^2 + A = 0 \)
• C. \( (A + I)(A + 2I) = 0 \)
• D. \( A^3 + 3A^2 - 2A = 0 \)

Hint:

Use the Cayley-Hamilton theorem to find polynomial relations for matrices. It avoids tedious direct multiplication and is especially effective for 2×2 and 3×3 matrices.

Answer 1

The Correct Option is D

Step 1: Use the Cayley-Hamilton Theorem.
The matrix \( A \) satisfies its own characteristic equation. First, we compute the characteristic polynomial of \( A \). \[ A = \begin{pmatrix} -3 & 2 \\ 1 & 0 \end{pmatrix} \] Step 2: Characteristic Polynomial
\[ \det(A - \lambda I) = \begin{vmatrix} -3 - \lambda & 2 \\ 1 & -\lambda \end{vmatrix} = (-3 - \lambda)(-\lambda) - (2)(1) = \lambda(3 + \lambda) - 2 = \lambda^2 + 3\lambda - 2 \] So the characteristic equation is: \[ \lambda^2 + 3\lambda - 2 = 0 \] Step 3: Cayley-Hamilton Theorem
According to the Cayley-Hamilton theorem, the matrix \( A \) satisfies: \[ A^2 + 3A - 2I = 0 \quad \text{(Equation 1)} \] Step 4: Multiply Equation (1) by \( A \)
\[ A(A^2 + 3A - 2I) = 0 \Rightarrow A^3 + 3A^2 - 2A = 0 \] Step 5: Conclusion
Therefore, the correct relation is: \[ \boxed{A^3 + 3A^2 - 2A = 0} \]