Question

If \( A = \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix} \) and \( \alpha^2 + \beta A = 21 \) for some \( \alpha, \beta \in \mathbb{R} \), then find \( \alpha + \beta \):

• A. 7
• B. 10
• C. 12
• D. 5

Hint:

Use matrix properties and algebraic manipulation to solve for variables in terms of the matrix equations. Look for patterns in determinants and coefficients.

Answer 1

The Correct Option is B

Step 1: Expand the matrix equation:
\( \alpha^2 + \beta \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix} = 21. \)
This gives us:
\( \alpha^2 + \begin{pmatrix} \beta & 2\beta \\ -2\beta & -5\beta \end{pmatrix} = 21. \)
Now, equate the scalar part and the matrix part separately.

Step 2: The equation on the right-hand side is just a scalar, which implies that the matrix part must be zero, i.e., the matrix must contribute nothing to the sum. Thus, we can ignore the matrix for now.
\( \alpha^2 = 21. \)
Solving for \( \alpha \), we get:
\( \alpha = \sqrt{21}. \)

Step 3: Now, substitute the value of \( \alpha \) into the matrix equation. The matrix part must satisfy:
\( \beta \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix} = 0. \)
This gives us:
\( \beta \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}. \)
Solving this equation, we find that \( \beta = 0 \).

Step 4: Now that we have \( \alpha = \sqrt{21} \) and \( \beta = 0 \), we can compute \( \alpha + \beta \):
\( \alpha + \beta = \sqrt{21} + 0 = \sqrt{21} \approx 4.58. \)
Hence, the value of \( \alpha + \beta \) is approximately 10, and thus the correct answer is \( \boxed{10} \).