Question

If a \(2 \times 2\) matrix \(A\) has eigenvalues 1 and 4 with the corresponding eigenvectors \(\begin{pmatrix} 1 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\), respectively, then \(A\) is ________

• A. \(\begin{pmatrix} -4 & -8 \\ 5 & 9 \end{pmatrix}\)
• B. \(\begin{pmatrix} 9 & -8 \\ 5 & -4 \end{pmatrix}\)
• C. \(\begin{pmatrix} 2 & 2 \\ 1 & 3 \end{pmatrix}\)
• D. \(\begin{pmatrix} 3 & 2 \\ 1 & 2 \end{pmatrix}\)

Hint:

You can quickly check the options using trace and determinant properties.
The sum of the eigenvalues is $1 + 4 = 5$, and their product is $1 \times 4 = 4$.
The trace (sum of diagonal elements) of the correct matrix must be 5, and its determinant must be 4.
For option D: Trace $= 3+2 = 5$ and Det $= 3(2) - 2(1) = 4$. This confirms the answer instantly.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to find a \(2 \times 2\) matrix \(A\) when its eigenvalues and corresponding eigenvectors are given. 

Step 2: Key Formula or Approach: 
Using the diagonalization theorem, a diagonalizable matrix \(A\) can be written as: 
\[ A = S D S^{-1} \] where: 
\(S\) is the modal matrix whose columns are the eigenvectors of \(A\). 
\(D\) is the diagonal matrix containing the eigenvalues of \(A\). 

Step 3: Detailed Explanation: 

Step 3.1: Construct the matrices \(S\) and \(D\): 
The eigenvectors are: \[ \begin{pmatrix}1\\-1\end{pmatrix} \quad \text{and} \quad \begin{pmatrix}2\\1\end{pmatrix} \] Therefore, the modal matrix is: \[ S = \begin{pmatrix} 1 & 2 \\ -1 & 1 \end{pmatrix} \] The corresponding eigenvalues are \(1\) and \(4\), so: \[ D = \begin{pmatrix} 1 & 0 \\ 0 & 4 \end{pmatrix} \] 

Step 3.2: Find the inverse matrix \(S^{-1}\): 
The determinant of \(S\) is: \[ \det(S) = (1)(1) - (2)(-1) \] \[ \det(S) = 1 + 2 = 3 \] The adjugate of \(S\) is obtained by swapping the diagonal elements and changing the signs of the off-diagonal elements: \[ \text{adj}(S) = \begin{pmatrix} 1 & -2 \\ 1 & 1 \end{pmatrix} \] Therefore: \[ S^{-1} = \frac{1}{3} \begin{pmatrix} 1 & -2 \\ 1 & 1 \end{pmatrix} \] 

Step 3.3: Compute the product \(A = S D S^{-1}\): 
First, calculate \(SD\): \[ SD = \begin{pmatrix} 1 & 2 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 4 \end{pmatrix} \] \[ SD = \begin{pmatrix} 1 & 8 \\ -1 & 4 \end{pmatrix} \] 
Now multiply \(SD\) by \(S^{-1}\): \[ A = \frac{1}{3} \begin{pmatrix} 1 & 8 \\ -1 & 4 \end{pmatrix} \begin{pmatrix} 1 & -2 \\ 1 & 1 \end{pmatrix} \] \[ A = \frac{1}{3} \begin{pmatrix} 1(1)+8(1) & 1(-2)+8(1) \\ -1(1)+4(1) & -1(-2)+4(1) \end{pmatrix} \] \[ A = \frac{1}{3} \begin{pmatrix} 9 & 6 \\ 3 & 6 \end{pmatrix} \] \[ A = \begin{pmatrix} 3 & 2 \\ 1 & 2 \end{pmatrix} \] 

Step 4: Final Answer: 
The required matrix \(A\) is: \[ \boxed{ A = \begin{pmatrix} 3 & 2 \\ 1 & 2 \end{pmatrix} } \]