If [A = [ pmatrix 1 & 0 1 & 1 pmatrix ]] and [I = [ pmatrix 1 & 0 0 & 1 pmatrix ],] then w…

Question

If $$A = \left[ {\begin{pmatrix} 1 & 0  \
   1 & 1  \ \end{pmatrix} } \right]$$ and $$I = \left[ {\begin{pmatrix} 1 & 0  \
   0 & 1  \ \end{pmatrix} } \right],$$ then which one of the following holds for all $$n \ge 1,$$ by the principle of mathematical induction?

Accepted Answer

Given $$A = \left[ {\begin{pmatrix} 1 & 0  \
   1 & 1  \

\end{pmatrix} } \right]$$

$$\therefore$$ $$A \times A$$ = ${A^2}$ = $$\left[ {\begin{pmatrix} 1 & 0  \
   2 & 1  \

\end{pmatrix} } \right]$$

and ${A^3}$ = $${A^2} \times A$$ = $$\left[ {\begin{pmatrix} 1 & 0  \
   3 & 1  \

\end{pmatrix} } \right]$$

So we can say ${A^n}$ = $$\left[ {\begin{pmatrix} 1 & 0  \
   n & 1  \

\end{pmatrix} } \right]$$

Now $$nA - \left( {n - 1} \right){\rm I}$$

= $$\left[ {\begin{pmatrix} n & 0  \
   n & n  \

\end{pmatrix} } \right]$$ - $$\left[ {\begin{pmatrix} {n - 1} & 0  \
   0 & {n - 1}  \

\end{pmatrix} } \right]$$

= $$\left[ {\begin{pmatrix} 1 & 0  \
   n & 1  \

\end{pmatrix} } \right]$$ = ${A^n}$

$$\therefore$$ $${A^n} = nA - \left( {n - 1} \right){\rm I}$$

Solutions · 1