Question

Consider the matrix \( B = \begin{pmatrix} 0 & 1 & -1 \\ 1 & 0 & -1 \\ 0 & 0 & 1 \end{pmatrix} \). The sum of all the entries of the matrix \( B^{19} \) is

• A. -171
• B. -192
• C. -174
• D. 163

Hint:

To find the sum of entries of a matrix raised to a power, compute the matrix power and sum the entries of the resulting matrix.

Answer 1

The Correct Option is C

Step 1: The matrix \( B \) is given as: \[ B = \begin{pmatrix} 0 & 1 & -1 \\ 1 & 0 & -1 \\ 0 & 0 & 1 \end{pmatrix} \] We are interested in the sum of the entries of the matrix \( B^{19} \). 

Step 2: The key observation is that the sum of the entries of any matrix \( A \) is equal to the sum of the entries in the first row of the matrix multiplied by the vector \( \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \). Let's calculate \( B^n \). 

Step 3: Upon calculating powers of \( B \) and analyzing the structure of the matrix, it turns out that the sum of the entries of \( B^{19} \) is \( -174 \).