1. If the sum of all the elements of a \( 3 \times 3 \) scalar matrix is 9 , then the product of all its elements i \( \begin{array}{llll}\text { (a) } 0 & \text { (b) } 9 & \text { (c) } 27\end{array} \) 2. Let \( f: R, \rightarrow[-5, a c) \) be defined \( a s f(x)=9 x^{2}+6 x-5 \), where \( R \), is the set of all non-negative re.

The given scalar matrix is of the form \( \begin{pmatrix} x & 0 & 0 \\ 0 & x & 0 \\ 0 & 0 & x \end{pmatrix} \). Since the sum of its elements (3x) equals 9, we determine that \( x = 3 \). Thus, the product of all elements becomes \( 3 \times 3 \times 3 = 27\). So, the correct choice is (c) 27! Diving deeper into quadratic functions, like the one you've given, helps us understand parabolas and their properties. The function \( f(x) = 9x^2 + 6x - 5 \) opens upward since the quadratic coefficient (9) is positive. Analyzing the vertex can reveal its minimum or maximum value, which is especially useful if you're seeking to determine the range of \( f \) over non-negative values of \( x \). It's a great way to root out the complete behavior of such functions!