Let $$A = \left[ {\matrix{ i & { - i} \cr { - i} & i \cr } } \right],i = \sqrt { - 1} $$. Then, the system of linear equations $${A^8}\left[ {\matrix{ x \cr y \cr } } \right] = \left[ {\matrix{ 8 \cr {64} \cr } } \right]$$ has :
Solution
$$A = \left[ {\matrix{
i & { - i} \cr
{ - i} & i \cr
} } \right]$$<br><br>$${A^2} = \left[ {\matrix{
i & { - i} \cr
{ - i} & i \cr
} } \right]\left[ {\matrix{
i & { - i} \cr
{ - i} & i \cr
} } \right] = \left[ {\matrix{
{ - 2} & 2 \cr
2 & { - 2} \cr
} } \right] = 2\left[ {\matrix{
{ - 1} & 1 \cr
1 & { - 1} \cr
} } \right]$$<br><br>$${A^4} = 4\left[ {\matrix{
{ - 1} & 1 \cr
1 & { - 1} \cr
} } \right]\left[ {\matrix{
{ - 1} & 1 \cr
1 & { - 1} \cr
} } \right] = 4\left[ {\matrix{
2 & { - 2} \cr
{ - 2} & 2 \cr
} } \right] = 8\left[ {\matrix{
1 & { - 1} \cr
{ - 1} & 1 \cr
} } \right]$$<br><br>$${A^8} = 64\left[ {\matrix{
1 & { - 1} \cr
{ - 1} & 1 \cr
} } \right]\left[ {\matrix{
1 & { - 1} \cr
{ - 1} & 1 \cr
} } \right] = 64\left[ {\matrix{
2 & { - 2} \cr
{ - 2} & 2 \cr
} } \right] = 128\left[ {\matrix{
1 & { - 1} \cr
{ - 1} & 1 \cr
} } \right]$$<br><br>$$128\left[ {\matrix{
1 & { - 1} \cr
{ - 1} & 1 \cr
} } \right]\left[ {\matrix{
x \cr
y \cr
} } \right] = \left[ {\matrix{
8 \cr
{64} \cr
} } \right]$$<br><br>$$128\left[ {\matrix{
{x - y} \cr
{ - x + y} \cr
} } \right] = \left[ {\matrix{
8 \cr
{64} \cr
} } \right] $$
<br><br>$\Rightarrow 128(x - y) = 8$<br><br>$\Rightarrow x - y = {1 \over {16}}$ .... (1)<br><br>and $128( - x + y) = 64 \Rightarrow x - y = {{ - 1} \over 2}$ .... (2)<br><br>$\Rightarrow$ No solution (from eq. (1) & (2))
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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