Deck 11: Determinants

Question 1

Find the value of the second-order determinant: $\left|\begin{array}{ll}4 & 4 \\ 4 & 4\end{array}\right|$

Answer 1

0

Question 2

Find the value of the second-order determinant: $\left|\begin{array}{cc}-4 & 3 \\ 6 & -2\end{array}\right|$

Answer 2

-10

Question 3

Find the value of the second-order determinant: $\left|\begin{array}{cc}5 & -10 \\ 1 & -2\end{array}\right|$

Answer 3

0

Question 4

Find the value of the second-order determinant: $\left|\begin{array}{cc}-2 & 1 \\ 7 & -2\end{array}\right|$

Question 5

Find the value of the second-order determinant: $\left|\begin{array}{ll}x & y \\ w & z\end{array}\right|$

Question 6

Find the value of the second-order determinant: $\left|\begin{array}{cc}3 a & 5 \\ 2 a & -6\end{array}\right|$

Question 7

Find the value of the second-order determinant: $\left|\begin{array}{cc}5 & x^{2}-36 \\ 1 & (x-6)^{2}\end{array}\right|$

Question 8

Find the value of the second-order determinant: $\left|\begin{array}{cc}(x-2)^{2} & x^{2}+4 x+4 \\ x-2 & x+2\end{array}\right|$

Question 9

Find the value of the second-order determinant: $\left|\begin{array}{cc}-x^{2}+5 x & x \\ 6 & -2\end{array}\right|$

Question 10

Find the value of the second-order determinant: $\left|\begin{array}{ll}\frac{2}{5} & \frac{-1}{4} \\ \frac{4}{5} & \frac{3}{4}\end{array}\right|$

Question 11

Find the value of the second-order determinant: $\left|\begin{array}{cc}\frac{2}{3} & \frac{1}{3} \\ \frac{-1}{3} & \frac{-2}{3}\end{array}\right|$

Question 12

Find the value of the second-order determinant: $\left|\begin{array}{cc}3 & \frac{1}{5} \\ \frac{-10}{3} & \frac{2}{9}\end{array}\right|$

Question 13

Find the value of the second-order determinant: $\left|\begin{array}{cc}\frac{1}{5} & \frac{3}{5} \\ \frac{-6}{30} & \frac{9}{15}\end{array}\right|$

Question 14

Solve by determinants: $3 x+2 y=4$
$x-2 y=2$

Question 15

Solve by determinants: $11 x-5 y=3$
$2 x+6 y=4$

Question 16

Solve by determinants: $2 x-4 y=6$
$5 x-y=9$

Question 17

Solve by determinants: $3 x-4 y=-22$
$5 x+3 y=60$

Question 18

Solve by determinants: $x+6 y=13$
$4 x-3 y=-29$

Question 19

Solve by determinants: $2 x+4 y=16$
$3 x-2 y=-8$

Question 20

Solve by determinants: $-2 y-6=5 x$
$2 y+2 x-9=0$

Question 21

Solve by determinants: $2 x=3 y+18$
$y=5-3 x$

Question 22

Solve by determinants: $\frac{12}{x}+\frac{3}{y}=4$
$\frac{20}{x}-\frac{9}{y}=2$

Question 23

Solve by determinants: $\frac{6}{x}+\frac{4}{y}=3$
$\frac{-9}{x}+\frac{16}{y}=1$

Question 24

Solve by determinants: $\frac{x}{4}+\frac{3 y}{5}=1$
$\frac{x}{2}-\frac{2 y}{3}=2$

Question 25

Solve for $x$ and $y$ by determinants: $4 m x-2 n y=2$
$3 m x+5 n y=6$

Question 26

Solve for $x$ and $y$ by determinants: $(a+2) x+(a-3) y=5$
$(a-4) x+(a+6) y=5$

Question 27

A shipment of 10 calculators and 8 flash drives cost $\$ 246$ . Another shipment of 4 calculators and 10 flash drives cost $\$ 180$ . What does each item cost?

Question 28

Calculate the currents $I_{1}$ and $I_{2}: 3.00 I_{1}+5.00\left(I_{1}-I_{2}\right)=12.0$
$10.0 I_{2}+5.00\left(I_{2}-I_{1}\right)=6.00$

Question 29

Evaluate the determinant: $\left|\begin{array}{ccc}2 & 1 & -4 \\ -3 & 8 & 1 \\ 5 & 3 & 4\end{array}\right|$

Question 30

Evaluate the determinant: $\left|\begin{array}{ccc}3 & -1 & 1 \\ -1 & 2 & 4 \\ 4 & -2 & 5\end{array}\right|$

Question 31

Evaluate the determinant: $\left|\begin{array}{ccc}-1 & 2 & -4 \\ 1 & -5 & 6 \\ 3 & 0 & 7\end{array}\right|$

Question 32

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

Question 33

Evaluate the determinant: $\left|\begin{array}{lll}4 & 1 & 3 \\ 1 & 2 & 0 \\ 6 & 4 & 1\end{array}\right|$

Question 34

Evaluate the determinant: $\left|\begin{array}{ccc}1.5 & 1.8 & -2.1 \\ -1.9 & 2.2 & 1.6 \\ -2.3 & 2.0 & 1.7\end{array}\right|$

Question 35

Evaluate the determinant: $\left|\begin{array}{lll}2 & 1 & 3 \\ 0 & 2 & 0 \\ 6 & 5 & 7\end{array}\right|$

Question 36

Evaluate the determinant: $\left|\begin{array}{ccc}1.0 & 8.8 & -3.1 \\ 7.8 & 1.5 & 5.3 \\ -3.3 & 2.0 & 6.4\end{array}\right|$

Question 37

Evaluate the determinant: $\left|\begin{array}{ccc}0.3 & 1.2 & -0.5 \\ 0.4 & -1.4 & 1.3 \\ -0.2 & 1.3 & 2.2\end{array}\right|$

Question 38

Evaluate the determinant: $\left|\begin{array}{ccc}1.25 & 2.10 & 1.55 \\ 0.00 & -1.10 & 2.00 \\ 3.50 & 4.11 & 1.35\end{array}\right|$

Question 39

Solve by determinants: $x-2 y+z=3$
$\begin{array}{r}2 x+2 y-3 z=9 \\2 x-3 y+z=-9\end{array}$

Question 40

Solve by determinants: $x+2 y-2 z=54$
$\begin{gathered}3 x-y+4 z=-26 \\2 x+3 y+z=49\end{gathered}$

Question 41

Solve by determinants: $x+y-2 z=14$
$\begin{aligned}2 x-y+3 z & =-10 \\2 x+3 y-2 z & =8\end{aligned}$

Question 42

Solve by determinants: $x+y+z=6$
$\begin{aligned}-x+y & =-8 \\x-z & =1\end{aligned}$

Question 43

Solve by determinants: $x+2 y+z=5$
$\begin{array}{r}-2 x+y=-5 \\y-z=-6\end{array}$

Question 44

Solve by determinants: $3 x+2 y=6$
$\begin{aligned}& 3 y-4 z=7 \\& 2 x+3 z=7\end{aligned}$

Question 45

Solve by determinants: $5 x+2 z=1$
$\begin{aligned}& 3 y-2 z=4 \\& 2 x+3 y=-1\end{aligned}$

Question 46

Solve by determinants: $2 x-y=1$
$\begin{aligned}& 4 y-z=3 \\& x-3 z=-2\end{aligned}$

Question 47

Solve by determinants: $7 x+13 y-7 z=24$
$4 x-6 y+10 z=37$
$6 x-26 y+33 z=15$

Question 48

Solve by determinants: $3 x-5 y+14 z=-55$
$\begin{aligned}10 x+2 y-7 z & =43 \\2 x-y+z & =-10\end{aligned}$

Question 49

Solve by determinants: $2.5 x+13.2 y+4.2 z=4.4$
$\begin{array}{r}3.2 x+2.5 y-2.2 z=-2.7 \\1.5 x-2.4 y-3.5 z=-5.1\end{array}$

Question 50

Solve by determinants: $1.02 x-2.03 y=9.16$
$\begin{aligned}& -2.05 x+3.04 z=-4.17 \\& -1.04 y-2.05 z=-2.02\end{aligned}$

Question 51

Solve by determinants: $\frac{1}{x}+\frac{1}{y}-\frac{1}{z}=-2$
$\begin{aligned}& \frac{2}{x}-\frac{3}{y}+\frac{2}{z}=13 \\& \frac{1}{x}+\frac{2}{y}+\frac{3}{z}=9\end{aligned}$

Question 52

Solve by determinants: $\frac{1}{x}-\frac{2}{y}+\frac{1}{z}=3$
$\begin{aligned}& \frac{2}{x}+\frac{2}{y}-\frac{3}{z}=9 \\& \frac{2}{x}-\frac{3}{y}+\frac{1}{z}=-9\end{aligned}$

Question 53

Determine the coefficients $a, b$ , and $c$ which balance the chemical equation $a \mathrm{Fe}_{2} \mathrm{O}_{3}+b \mathrm{Al} \rightarrow c \mathrm{Al}_{2} \mathrm{O}_{3}+$ $10 \mathrm{Fe}$ . In other words, solve this system:
Fe: $\quad 2 a+0 b=10$
O: $\quad 3 a+0 b=3 c$
Al: $\quad 0 a+b=2 c$

Question 54

Determine the coefficients $a, b$ , and $c$ which balance the chemical equation $a \mathrm{C}_{4} \mathrm{H}_{10}+13 \mathrm{O}_{2} \rightarrow b \mathrm{H}_{2} \mathrm{O}+$ $c \mathrm{CO}_{2}$ . In other words, solve this system:
C: $\quad 4 a=0 b+c$
$\mathrm{H}: \quad 10 a=2 b+0 c$
O: $\quad 26=b+2 c$

Question 55

Kirchhoff's rules were applied to a three-loop network. Solve for currents $l_{1}, l_{2}$ , and $l_{3}$ in this resulting system of equations:
$76.0 I_{1}-25.0 I_{2}-50.0 I_{3}=10.0$
$-25.0 l_{1}+56.0 I_{2}-1.00 I_{3}=0.00$
$-50.0 l_{1}-1.0 I_{2}+106 l_{3}=0.00$

Question 56

Evaluate the determinant: $\left|\begin{array}{cccc}1 & 0 & 2 & 3 \\ -3 & -1 & 0 & -2 \\ 2 & 3 & 1 & 0 \\ 0 & 2 & 3 & 1\end{array}\right|$

Question 57

Evaluate the determinant: $\left|\begin{array}{cccc}3 & 1 & 4 & 0 \\ -2 & 3 & -1 & 6 \\ 0 & 1 & 3 & -1 \\ 0 & 4 & -5 & 2\end{array}\right|$

Question 58

Evaluate the determinant: $\left|\begin{array}{cccc}2 & -1 & 7 & 1 \\ 0 & 4 & -3 & 4 \\ -1 & 6 & -2 & 1 \\ 3 & 1 & 0 & -1\end{array}\right|$

Question 59

Evaluate the determinant: $\left|\begin{array}{cccc}1 & 4 & -5 & 5 \\ 3 & 1 & 0 & 2 \\ -1 & 2 & 0 & 2 \\ 2 & -4 & 6 & 4\end{array}\right|$

Question 60

Evaluate the determinant: $\left|\begin{array}{cccc}2 & 5 & 3 & 4 \\ 0 & 1 & 2 & -1 \\ 4 & 3 & 2 & 2 \\ 7 & 3 & 5 & 1\end{array}\right|$

Question 61

Evaluate the determinant: $\left|\begin{array}{cccc}1 & -4 & 2 & 12 \\ 3 & 1 & 0 & 2 \\ 0 & 2 & 0 & 6 \\ 6 & 3 & 0 & 10\end{array}\right|$

Question 62

Evaluate the determinant: $\left|\begin{array}{cccc}0 & 5 & 2 & 4 \\ 0 & 1 & 2 & -8 \\ 4 & 0 & 12 & 2 \\ 7 & 1 & 1 & 1\end{array}\right|$

Question 63

Evaluate the determinant: $\left|\begin{array}{ccccc}2 & -1 & 3 & 2 & 5 \\ 5 & 3 & 6 & -4 & 1 \\ 3 & 0 & 6 & 2 & 4 \\ 1 & 4 & 4 & 5 & 2 \\ 2 & 4 & -1 & 3 & 6\end{array}\right|$

Question 64

Evaluate the determinant: $\left|\begin{array}{ccccc}2 & 3 & -4 & -1 & 0 \\ 7 & 3 & 2 & 2 & -2 \\ 1 & -2 & 3 & 1 & 0 \\ -1 & -2 & -3 & 1 & 1 \\ 1 & 2 & -4 & 5 & -1\end{array}\right|$

Question 65

Solve by determinants: $3 x+2 y-5 z+w=2$
$\begin{aligned}& x-2 y+z-w=2 \\& x+y+z+2 w=2 \\& x-y-z-3 w=-1\end{aligned}$

Question 66

Solve by determinants: $x+2 y+z+w=2$
$\begin{aligned}x+3 y+2 z+w & =0 \\2 x-y+2 z-3 w & =-1 \\x+y-4 z-w & =-14\end{aligned}$

Question 67

Solve by determinants: $2 x+5 y-3 z-4 w=20$
$\begin{aligned}7 x-3 y+z-2 w & =5 \\x-3 y-z-w & =-10 \\x+3 y+4 z+5 w & =3\end{aligned}$

Question 68

Solve by determinants: $3 x+3 z=6$
$\begin{aligned}4 y+2 z & =-12 \\2 x+6 y+4 z & =9 \\2 x+4 y+4 w & =-8\end{aligned}$

Question 69

Solve by determinants: $2 x+y-2 z+w=-14$
$\begin{array}{r}x-2 y+z=-2 \\3 x+y-2 w=-8 \\3 y+3 z-w=25\end{array}$

Question 70

Solve by determinants: $2 x+2 y-3 z=11$
$\begin{array}{r}3 y+2 z+w=10 \\3 x-4 y+2 w=-4 \\z+3 w-v=14 \\3 y+2 w=23\end{array}$

Question 71

Solve by determinants: $x+2 z-w+u=1$
$3 y+z+2 w+u=-2$
$\begin{array}{r}-x-2 y-2 z+3 w+u=15 \\x+3 z-w+u=2.5\end{array}$
$x+y+z+w+u=1$

Question 72

Solve by determinants: $2 a+5 b-3 c-4 d+e=2$
$\begin{aligned}7 a-3 b+d-2 e & =2 \\a-3 c-d-e & =-6 \\3 b+4 c+5 d-2 e & =-11 \\a-3 b-4 c+d & =-6\end{aligned}$

Question 73

Solve by determinants: $a+b-c-d+\mathrm{e}=-3$
$\begin{aligned}& a-b+d-e=3 \\& a-c-d-e=2 \\& b+c+d-e=4 \\& a-b-c+d=2\end{aligned}$

Question 74

Kirchhoff's rules were applied to a four-loop network. Solve for currents $I_{1}, l_{2}, l_{3}$ , and $I_{4}$ in this resulting system of equations:
$-6.0 I_{1}+2.0 I_{2}+4.0 I_{4}=-12$
$4.0 I_{1}+4.0 I_{2}-14 I_{4}=0$
$2.0 I_{1}-8.0 I_{2}+20 I_{3}+4.0 I_{4}=0$
$2.0 I_{2}-2.0 I_{3}=12$

Question 75

Kirchhoff's rules were applied to a four-loop network. Solve for currents $I_{1}, I_{2}, l_{3}$ , and $I_{4}$ in this resulting system of equations:
$-2 I_{1}+3 I_{2}+2 I_{3}=-18$
$\begin{aligned}3 I_{1}+3 I_{3}-4 I_{4} & =-13 \\I_{1}+5 I_{2}+5 I_{4} & =0 \\-10 I_{1}+I_{2}-5 I_{3}-8 I_{4} & =5\end{aligned}$
Find the loop currents.