Question

If A = \left[ {\begin{array}{*{20}{c}} 1&2 \\\ { - 1}&3 \end{array}} \right] and ${A^6} = KA - 205I$ then
(A) $K =$ $11$
(B) $K =$ $22$
(C) $K =$ $33$
(D) $K =$ $44$

Answer 1

Find ${A^6}$ in multiple steps, i.e., ${A^2} = A \times A$, ${A^3} = {A^2} \times A$ and ${A^6} = {A^3} \times {A^3}$. Then use the given relation ${A^6} = KA - 205I$ to find the value of $K$.

Complete step by step solution:
Given, A = \left[ {\begin{array}{*{20}{c}} 1&2 \\\ { - 1}&3 \end{array}} \right]
We have to calculate ${A^6}$. For this, we find firstly ${A^2}$ and ${A^3}$.

1&2 \\\ { - 1}&3 \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} 1&2 \\\ { - 1}&3 \end{array}} \right]$$ $ \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}} {1 \times 1 + 2 \times - 1}&{1 \times 2 + 2 \times 3} \\\ { - 1 \times 1 + 3 \times - 1}&{ - 1 \times 2 + 3 \times 3} \end{array}} \right]$ $ \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}} {1 - 2}&{2 + 6} \\\ { - 1 - 3}&{ - 2 + 9} \end{array}} \right]$ $ \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}} { - 1}&8 \\\ { - 4}&7 \end{array}} \right]$ And $${A^3} = {A^2} \times A = \left[ {\begin{array}{*{20}{c}} { - 1}&8 \\\ { - 4}&7 \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} 1&2 \\\ { - 1}&3 \end{array}} \right]$$ $ \Rightarrow {A^3} = \left[ {\begin{array}{*{20}{c}} { - 1 \times 1 + 8 \times - 1}&{ - 1 \times 2 + 8 \times 3} \\\ { - 4 \times 1 + 7 \times - 1}&{ - 4 \times 2 + 7 \times 3} \end{array}} \right]$ $ \Rightarrow {A^3} = \left[ {\begin{array}{*{20}{c}} { - 1 - 8}&{ - 2 + 24} \\\ { - 4 - 7}&{ - 8 + 21} \end{array}} \right]$ $ \Rightarrow {A^3} = \left[ {\begin{array}{*{20}{c}} { - 9}&{22} \\\ { - 11}&{13} \end{array}} \right]$ Now, $${A^6} = {A^3} \times {A^3} = \left[ {\begin{array}{*{20}{c}} { - 9}&{22} \\\ { - 11}&{13} \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} { - 9}&{22} \\\ { - 11}&{13} \end{array}} \right]$$ $ \Rightarrow {A^6} = \left[ {\begin{array}{*{20}{c}} { - 9 \times - 9 + 22 \times - 11}&{ - 9 \times 22 + 22 \times 13} \\\ { - 11 \times - 9 + 13 \times - 11}&{ - 11 \times 22 + 13 \times 13} \end{array}} \right]$ $ \Rightarrow {A^6} = \left[ {\begin{array}{*{20}{c}} {81 - 242}&{ - 198 + 286} \\\ {99 - 143}&{ - 242 + 169} \end{array}} \right]$ $ \Rightarrow {A^6} = \left[ {\begin{array}{*{20}{c}} { - 161}&{88} \\\ { - 44}&{ - 73} \end{array}} \right]$ We have ${A^6} = KA - 205I$ $ \Rightarrow $ ${A^6} + 205I = KA$ ….. (1) Substitute the value of ${A^6}$, $I$ and $A$ in (1); $\left[ {\begin{array}{*{20}{c}} { - 161}&{88} \\\ { - 44}&{ - 73} \end{array}} \right] + 205\left[ {\begin{array}{*{20}{c}} 1&0 \\\ 0&1 \end{array}} \right] = K\left[ {\begin{array}{*{20}{c}} 1&2 \\\ { - 1}&3 \end{array}} \right]$ $ \Rightarrow \left[ {\begin{array}{*{20}{c}} { - 161}&{88} \\\ { - 44}&{ - 73} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {205}&0 \\\ 0&{205} \end{array}} \right] = K\left[ {\begin{array}{*{20}{c}} 1&2 \\\ { - 1}&3 \end{array}} \right]$ $ \Rightarrow \left[ {\begin{array}{*{20}{c}} { - 161 + 205}&{88 + 0} \\\ { - 44 + 0}&{ - 73 + 205} \end{array}} \right] = K\left[ {\begin{array}{*{20}{c}} 1&2 \\\ { - 1}&3 \end{array}} \right]$ $ \Rightarrow \left[ {\begin{array}{*{20}{c}} {44}&{88} \\\ { - 44}&{132} \end{array}} \right] = K\left[ {\begin{array}{*{20}{c}} 1&2 \\\ { - 1}&3 \end{array}} \right]$ $ \Rightarrow 44\left[ {\begin{array}{*{20}{c}} 1&2 \\\ { - 1}&3 \end{array}} \right] = K\left[ {\begin{array}{*{20}{c}} 1&2 \\\ { - 1}&3 \end{array}} \right]$ On comparing both sides, we get- $$ \Rightarrow K = 44$$ **Hence, option (D) is the correct answer.** **Note:** Here, the symbol $I$ is used for identity matrices that have an order $n \times n$. The entries on the diagonal from the upper left to the bottom right are all $1's$ and all other entries are $0$. For ex- ${I_2} = \left[ {\begin{array}{*{20}{c}} 1&0 \\\ 0&1 \end{array}} \right]$ , ${I_3} = \left[ {\begin{array}{*{20}{c}} 1&0&0 \\\ 0&1&0 \\\ 0&0&1 \end{array}} \right]$ The identity matrix plays a similar role in operations with matrices as the number $1$ plays in operations with real numbers.