Question

Let
 \(x = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\) and \(A = \begin{bmatrix} -1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1 \end{bmatrix}\)
For k ∈ N, if X’A^kX = 33, then k is equal to ____ .

Answer 1

Correct Answer: 10

Given that, 
\(A = \begin{bmatrix} -1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1 \end{bmatrix}\)
\(A^2 = \begin{bmatrix} 1 & 0 & 6 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\)
\(A^4 = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix}\)
\(A^k = \begin{bmatrix} 1 & 0 & 3k \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\)
So, \(X^′A^kX= \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 0 & 3k \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\)
\(⇒X^′A^kX=[3k+3]\)
⇒ [3k + 3] = 33 (here it shall be [33] as matrix can’t be equal to a scalar)
i.e. [3k + 3] = 33
3k + 3 = [33] ⇒ k = 10
If k is odd and apply above process, we don’t get odd value of k
∴ k = 10
So, the answer is 10.