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’AkX = 33, then k is equal to ____ .

Answer 1

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.