answer:The inverse of the matrix is: left( begin{array}{ccc} frac{9}{7} & -frac{8}{7} & -frac{4}{7} -frac{4}{7} & frac{2}{7} & frac{1}{7} frac{20}{7} & -frac{17}{7} & -frac{12}{7} end{array} right) I have standardized the formatting, ensured that LaTeX is properly rendered, and made the question and answer more concise. The content is already complete and accurate, so no further modifications are needed.

answer:According to some historians, the caste system in India originated with the Aryans, nomadic herders from Central Asia who migrated into the Indian subcontinent by 1500 BCE. The Aryans established a caste system with four main groups of people: brahmins (priests), kshatriyas (warriors and aristocrats), vaishyas (peasants and merchants), and shudras (serfs).

answer:To complete the square, we need to add and subtract the square of half the coefficient of x. In this case, half of frac{23}{2} is frac{23}{4}, so we need to add and subtract left(frac{23}{4}right)^2 = frac{529}{16}. -8 x^2 + frac{23 x}{2} - 6 = -8 left(x^2 - frac{23 x}{8} + frac{529}{256}right) + frac{529}{16} - 6 = -8 left(x - frac{23}{32}right)^2 + frac{529}{16} - frac{96}{16} = -8 left(x - frac{23}{32}right)^2 - frac{239}{128} The answer is -8 left(x - frac{23}{32}right)^2 - frac{239}{128}

answer:Yes, we can use the following method: Let hat a represent the two's complement number a with its topmost bit flipped and the result interpreted as an unsigned number. Then, we have: acdot b = (hat a - 2^{n-1})cdot(hat b - 2^{m-1}) = hat a cdot hat b - 2^{m-1} hat a - 2^{n-1} hat b + 2^{n+m-2} Here, we can compute hat a cdot hat b using unsigned multipliers and then subtract the other terms to obtain the signed product. This approach eliminates the need for sign extension during partial product summation.