(Solved): (1) Use Cramer's rule to solve (if possible) the system of linear equations \[ \begin{aligned} 14 ...

(1) Use Cramer's rule to solve (if possible) the system of linear equations \[ \begin{aligned} 14 x_{1}-21 x_{2}-7 x_{3} & =-21 \\ -4 x_{1}+2 x_{2}-2 x_{3} & =2 \\ 56 x_{1}-21 x_{2}+7 x_{3} & =7 \end{aligned} \] (2) What are the values of \( h \) and \( k \) for which the matrix below is not invertible? \[ A=\left[\begin{array}{ccc} 1 & 1 & 1 \\ -1 & 0 & k \\ -1 & h & -1 \end{array}\right] \] (3) \( \left({ }^{* * *}\right. \) You can use Python to aid with your calculations \( { }^{* * *} \) ) If the matrices \( A \) and \( B \) are such that \( A=\left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right] \) and \( A B=\left[\begin{array}{lll}2 & 3 & 1 \\ 3 & 5 & 2\end{array}\right] \), then what is the matrix \( B \) ? (4) \( \left(^{* * *}\right. \) This problem is simple. Don't overthink it!***) Find all values of \( a \) for which the system \( A \overrightarrow{\boldsymbol{x}}=\overrightarrow{\mathbf{0}} \) has a nontrivial solution where \( A=\left[\begin{array}{ccc}1 & -1 & 2 \\ -1 & a & 2 \\ 3 & -3 & a\end{array}\right] \) by answering the following questions (a) Form the augmented matrix of the given system and use Gauss-Jordan elimination to show that an equivalent system of \( A \vec{x}=\vec{b} \) is \[ \begin{array}{r} x_{1}-x_{2}+2 x_{3}=0 \\ (a-1) x_{2}=0 \\ (a-6) x_{3}=0 \end{array} \] (b) What are the values of \( a \) such that the system has nontrivial solutions