Given matrices $A = (3512)$ and $B = (2 - 5 - 1 3)$ , find the product matrix $A B$ .

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If $f (x) = \frac{2 x + 1}{x - 3}$ for $x \neq = 3$ , determine the inverse function $f^{- 1} (x)$ .
The sum of an infinite geometric series with first term $a = 12$ is $S = 16$ . Find the common ratio $r$ of the series.
If $f (x) = 3 x - 2$ and $(g \circ f) (x) = 9 x^{2} - 12 x + 7$ , determine the function $g (x)$ .
For which value of $k$ does the matrix $M = (k 12 3 k)$ fail to be invertible (singular)?
In an arithmetic progression, the 4th term is 17 and the 10th term is 41. Find the sum of the first 20 terms ( $S_{20}$ ).
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