91Ó°ÊÓ

Evaluate: $$ \left|\begin{array}{rrrr} 2 & 0 & 1 & 3 \\ 3 & 1 & 0 & 4 \\ 1 & -1 & 2 & 3 \\ 2 & 2 & 1 & 0 \end{array}\right| $$

Evaluate: $$ \left|\begin{array}{llll} 3 & 4 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 4 & 2 \end{array}\right| $$

Evaluate: $$ \left|\begin{array}{rrrr} 1 & 1 & 0 & 0 \\ 3 & 2 & 2 & -3 \\ 2 & 1 & -1 & 2 \\ 5 & 3 & 1 & -1 \end{array}\right| $$

Evaluate: $$ \left|\begin{array}{rrrr} -2 & 6 & 17 & -5 \\ 0 & 3 & 22 & -17 \\ 0 & 0 & 4 & 12 \\ 0 & 0 & 0 & -6 \end{array}\right| $$

Use Cramer's rule to solve the systems of equations: \(3 x-2 y-2 z=0\) \(x+y-z=0\) \(2 x+2 y+z=0\)

Use Cramer's rule to solve the systems of equations: \(w+2 x+3 y+z=5\) \(2 w+x+y+z=3\) \(w+2 x+y=4\) \(x+y+2 z=0\)

(i) Show that the following equations have no solution unless \(k=3\), (ii) solve for this ralue of \(k\). $$ \begin{array}{r} 2 x-y+z=2 \\ 3 x+y-2 z=1 \\ x-3 v+4 z=k \end{array} $$

(i) Find \(k\) for which the following equations have a nontrivial solution, (ii) solve for this value of \(k\). $$ \begin{aligned} &k x+5 y+3 z=0 \\ &5 x+y-z=0 \\ &k x+2 y+z=0 \end{aligned} $$

Find (i) the values of \(\lambda\) for which the following systems of equations have nontrivial solutions, (ii) the solutions for these values of \(\lambda\). \begin{array}{r} 2 x+y=\lambda x \\ x+2 y=\lambda y \end{array}

Find (i) the values of \(\lambda\) for which the following systems of equations have nontrivial solutions, (ii) the solutions for these values of \(\lambda\). \(3 x+y=\lambda x\) \(x+3 y+z=\lambda y\) \(y+3 z=\lambda z\)