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beks73 [17]
3 years ago
5

Calculate the value of the following determinants: | 1 -1 3 2 5 0 -3 1 2 | and | -1 -8 2 9 1 0 4 1 2 |

Mathematics
1 answer:
vitfil [10]3 years ago
6 0

Answer:

1) The determinant = 65

2) The determinant = 152

Step-by-step explanation:

Let us show how to find the determinant of a matrix

You can find the determinant of this Matrix  \left[\begin{array}{ccc}a&b&c\\d&e&f\\g&m&n\end{array}\right]

by using this rule

Determinant = a(en - fm) - b(dn - fg) + c(dm - eg)

Let us use this rule with the given matrices

1)

 \left[\begin{array}{ccc}1&-1&3\\2&5&0\\-3&1&2\end{array}\right]

The determinand = 1[(5)(2) - (0)(1)] - (-1)[(2)(2) - (0)(-3)] + 3[(2)(1) - 5(-3)]

= 1[10 - 0] - (-1)[4 - 0] + 3[2 - (-15)]

= 1[10] + 1[4] + 3[2+15]

= 10 + 4 + 3[17]

= 10 + 4 + 51

= 65

The determinant = 65

Let us do the second one

2)

 \left[\begin{array}{ccc}-1&-8&2\\9&1&0\\4&1&2\end{array}\right]

The determinand = -1[(1)(2) - (0)(1)] - (-8)[(9)(2) - (0)(4)] + 2[(9)(1) - 1(4)]

= -1[2 - 0] - (-8)[18 - 0] + 2[9 - 4]

= -1[2] + 8[18] + 2[5]

= -2 + 144 + 10

= 152

The determinant = 152

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Answer:

The northern lighthouse is approximately 24.4\; \rm mi closer to the boat than the southern lighthouse.

Step-by-step explanation:

Refer to the diagram attached. Denote the northern lighthouse as \rm N, the southern lighthouse as \rm S, and the boat as \rm B. These three points would form a triangle.

It is given that two of the angles of this triangle measure 40^{\circ} (northern lighthouse, \angle {\rm N}) and 21^{\circ} (southern lighthouse \angle {\rm S}), respectively. The three angles of any triangle add up to 180^{\circ}. Therefore, the third angle of this triangle would measure 180^{\circ} - (40^{\circ} + 21^{\circ}) = 119^{\circ} (boat \angle {\rm B}.)

It is also given that the length between the two lighthouses (length of \rm NS) is 75\; \rm mi.

By the law of sine, the length of a side in a given triangle would be proportional to the angle opposite to that side. For example, in the triangle in this question, \angle {\rm B} is opposite to side \rm NS, whereas \angle {\rm S} is opposite to side {\rm NB}. Therefore:

\begin{aligned} \frac{\text{length of NS}}{\sin(\angle {\rm B})} = \frac{\text{length of NB}}{\sin(\angle {\rm S})} \end{aligned}.

Substitute in the known measurements:

\begin{aligned} \frac{75\; \rm mi}{\sin(119^{\circ})} = \frac{\text{length of NB}}{\sin(21^{\circ})} \end{aligned}.

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\begin{aligned} & \text{length of NB} \\ =\; & (75\; \rm mi) \times \frac{\sin(21^{\circ})}{\sin(119^{\circ})} \\ \approx\; & 30.73\; \rm mi\end{aligned}.

(Round to at least one more decimal places than the values in the choices.)

Likewise, with \angle {\rm N} is opposite to side {\rm SB}, the following would also hold:

\begin{aligned} \frac{\text{length of NS}}{\sin(\angle {\rm B})} = \frac{\text{length of SB}}{\sin(\angle {\rm N})} \end{aligned}.

\begin{aligned} \frac{75\; \rm mi}{\sin(119^{\circ})} = \frac{\text{length of SB}}{\sin(40^{\circ})} \end{aligned}.

\begin{aligned} & \text{length of SB} \\ =\; & (75\; \rm mi) \times \frac{\sin(40^{\circ})}{\sin(119^{\circ})} \\ \approx\; & 55.12\; \rm mi\end{aligned}.

In other words, the distance between the northern lighthouse and the boat is approximately 30.73\; \rm mi, whereas the distance between the southern lighthouse and the boat is approximately 55.12\; \rm mi. Hence the conclusion.

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<h3>Next number of the given sequence</h3>

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Thus, the first missing number in the sequence is 35, and the last missing number is 77.

Learn more about sequence here: brainly.com/question/6561461

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