Question

If A = \(\begin{bmatrix} 0 & 3\\ 0 & 0  \end{bmatrix}\)and f(x) = x+x²+x³+.....+x²⁰²³, then f(A)+I =

• A. \(\begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}\)
• B. \(\begin{bmatrix} 1 & 3\\ 0 & 0 \end{bmatrix}\)
• C. \(\begin{bmatrix} 1 & 3\\ 0 & 1 \end{bmatrix}\)
• D. \(\begin{bmatrix} 1 & 3\\ 1 & 1 \end{bmatrix}\)

Answer 1

The Correct Option is C

To solve the problem, we are given a matrix \( A = \begin{bmatrix} 0 & 3 \\ 0 & 0 \end{bmatrix} \) and a polynomial function \( f(x) = x + x^2 + x^3 + \cdots + x^{2023} \). We are to compute \( f(A) + I \), where \( I \) is the identity matrix.

1. Understanding the Polynomial Function:
The given polynomial is a geometric progression:

\( f(x) = x + x^2 + x^3 + \cdots + x^{2023} \)
This can be expressed as:
\( f(x) = \frac{x(x^{2023} - 1)}{x - 1} \) for \( x \ne 1 \), but since we are plugging in a matrix, we will analyze \( f(A) \) directly using matrix powers.

2. Powers of Matrix \( A \):
Given \( A = \begin{bmatrix} 0 & 3 \\ 0 & 0 \end{bmatrix} \), we compute its powers:

\( A^1 = \begin{bmatrix} 0 & 3 \\ 0 & 0 \end{bmatrix} \)
\( A^2 = A \cdot A = \begin{bmatrix} 0 & 3 \\ 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} 0 & 3 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
So, \( A^n = 0 \) for all \( n \geq 2 \)

3. Evaluate \( f(A) \):
Since \( A^2 = 0 \), we only consider the first term:
\( f(A) = A + A^2 + A^3 + \cdots + A^{2023} = A + 0 + 0 + \cdots + 0 = A \)

4. Add the Identity Matrix:
Now compute \( f(A) + I \):
\( f(A) + I = A + I = \begin{bmatrix} 0 & 3 \\ 0 & 0 \end{bmatrix} + \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \)

Final Answer:
The matrix \( f(A) + I \) is \( \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \)