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Symmetric Matrix

Let A=[ 1 -1 ...

Question

Let $A = ⎛ ⎜ ⎝ \begin{matrix} 1 & - 1 & 0 0 & 1 & - 1 0 & 0 & 1 \end{matrix} ⎞ ⎟ ⎠$ and $B = 7 A^{20} - 20 A^{7} + 2 I,$ where $I$ is an identity matrix of order $3 \times 3$ . If $B = [b_{i j}],$ then $b_{13}$ is equal to

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Solution

Let $A = I + P$ where $P = ⎡ ⎢ ⎣ \begin{matrix} 0 & - 1 & 0 0 & 0 & - 1 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦$
$P^{2} = ⎡ ⎢ ⎣ \begin{matrix} 0 & 0 & 1 0 & 0 & 0 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦$ and $P^{n} = 0$ for $n \geq 3$
So, $A^{n} = (I + P)^{n} = I +^{n} C_{1} \cdot P +^{n} C_{2} \cdot P^{2} + \dots + P^{n}$
So, $(b_{13})_{n} = 0 + 0 +^{n} C_{2} \cdot 1 =^{n} C_{2}$

Now $b_{13}$ element of $7 \cdot A^{20} - 20 \cdot A^{7} + 2 I$
$= 7 (^{20} C_{2}) - 20 (^{7} C_{2}) + 0$
$= 7 \times 190 - 20 \times 21$
$= 70 [19 - 6] = 910$

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