Problem 1
Use Cramer's rule to solve the following equation systems: \((a) 3 x_{1}-2 x_{2}=6\) \(2 x_{1}+x_{2}=11\) \(\langle b\rangle-x_{1}+3 x_{2}=-3\) \(4 x_{1}-x_{2}=12\) (c) \(8 x_{1}-7 x_{2}=9\) \(x_{1}+x_{2}=3\) \((d) 5 x_{1}+9 x_{2}=14\) \(7 x_{1}-3 x_{2}=4\)
Problem 1
Evaluate the following determinants: \((a) \left|\begin{array}{lll}8 & 1 & 3 \\ 4 & 0 & 1 \\ 6 & 0 & 3\end{array}\right|\) \((b) \left|\begin{array}{lll}1 & 2 & 3 \\ 4 & 7 & 5 \\\ 3 & 6 & 9\end{array}\right|\) \((c) \left|\begin{array}{lll}4 & 0 & 2 \\ 6 & 0 & 3 \\ 8 & 2 & 3\end{array}\right|\) \((d) \left|\begin{array}{rrr}1 & 1 & 4 \\ 8 & 11 & -2 \\ 0 & 4 & 7\end{array}\right|\) \((e) \left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) \((f) \left|\begin{array}{rrr}x & 5 & 0 \\ 3 & y & 2 \\ 9 & -1 & 8\end{array}\right|\)
Problem 1
Use the determinant \(\left|\begin{array}{ccc}4 & 0 & -1 \\ 2 & 1 & -7 \\ 3 & 3 & 9\end{array}\right|\) to verify the first four properties of determinants.
Problem 2
Find the inverse of each of the following matrices: \((a) A=\left[\begin{array}{ll}5 & 2 \\ 0 & 1\end{array}\right]\) (b) \(B=\left[\begin{array}{rr}-1 & 0 \\ 9 & 2\end{array}\right]\) \((c) C=\left[\begin{array}{rr}3 & 7 \\ 3 & -1\end{array}\right]\) \((d) D=\left[\begin{array}{ll}7 & 6 \\ 0 & 3\end{array}\right]\)
Problem 2
Determine the signs to be attached to the relevant minors in order to get the following cofactors of a determinant: $$\left|C_{13}|,| C_{231},\left|C_{33}\right|,\left|C_{41}\right|, \text { and }\left|C_{34} |\right.\right.$$
Problem 2
Let \(p\) be the statement "a geometric figure is a square," and let \(q\) be as follows: (a) it has four sides. (b) It has four equal sides. (c) It has four equal sides each perpendicular to the adjacent one. Which is true for each case: \(p \Rightarrow q, p \Leftarrow q,\) or \(p \Leftrightarrow q ?\)
Problem 3
Given \(\left|\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right|\) find the minors and cofactors of the elements \(a, b,\) and \(f\)
Problem 3
Which properties of determinants enable us to write the following? \((a)\left|\begin{array}{rr}9 & 27\\\18 & 56\end{array}\right|=\left|\begin{array}{ll}9 & 18 \\ 0 & 2\end{array}\right|\) \((b)\left|\begin{array}{rr}9 & 27 \\ 4 & 2\end{array}\right|=18\left|\begin{array}{ll}1 & 3 \\ 2 & 1\end{array}\right|\)
Problem 3
Are the rows linearly independent in each of the following? \((a)\left[\begin{array}{rr}24 & 8 \\ 9 & -3\end{array}\right]\) (b) \(\left[\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right]\) \((c)\left[\begin{array}{ll}0 & 4 \\ 3 & 2\end{array}\right]\) \((d)\left[\begin{array}{rr}-1 & 5 \\ 2 & -10\end{array}\right]\)
Problem 4
Find the inverse of each of the following matrices. (a) \(E=\left[\begin{array}{rrr}4 & -2 & 1 \\ 7 & 3 & 0 \\ 2 & 0 & 1\end{array}\right]\) (c) \(G=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]\) (b) \(F=\left[\begin{array}{rrr}1 & -1 & 2 \\ 1 & 0 & 3 \\ 4 & 0 & 2\end{array}\right]\) \((d) H=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\)
