Question

Let \(A = \begin{bmatrix} 0 & \alpha \\ 0 & 0 \end{bmatrix}\) and \((A + I)^{50} - 50A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) then the value of \(a + b + c + d\) is

• A. 2
• B. 1
• C. 4
• D. None of these

Hint:

For nilpotent matrices, binomial expansion truncates.

Answer 1

The Correct Option is A

Step 1: Formula / Definition}
\[ A^2 = \begin{bmatrix} 0 & \alpha \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & \alpha \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]
Step 2: Calculation / Simplification}
Since \(A^2 = 0\), \(A^k = 0\) for \(k \geq 2\)
\((A + I)^{50} = I + 50A\) (by binomial expansion)
\((A + I)^{50} - 50A = I + 50A - 50A = I\)
\(\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)
\(a=1, b=0, c=0, d=1\)
\(a+b+c+d = 2\)
Step 3: Final Answer
\[ 2 \]