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Chapter 9: Problem 9

Find the value of each determinant. $$\left|\begin{array}{rr} -7 & 0 \\ 3 & 0 \end{array}\right|$$

Short Answer

Expert verified
The determinant is 0.

Step by step solution

01

- Understand the Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is calculated using the formula: \(ad - bc\).
02

- Identify the Elements of the Matrix

In the given matrix \(\begin{bmatrix} -7 & 0 \ 3 & 0 \end{bmatrix}\), we identify \(a = -7\), \(b = 0\), \(c = 3\), and \(d = 0\).
03

- Apply the Determinant Formula

Plug in the values into the formula: \((-7) \times 0 - (0) \times 3\).
04

- Simplify the Expression

Calculate the expression: \((-7) \times 0 = 0\) and \(0 \times 3 = 0\). So, the determinant is \(0 - 0 = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

2x2 Matrix
A 2x2 matrix is a simple square arrangement of numbers with two rows and two columns. For instance, the matrix in the exercise \(\begin{bmatrix} -7 & 0 \ 3 & 0 \ \end{bmatrix}\) is a 2x2 matrix because it contains exactly four elements. Understanding the structure and notation of matrices is crucial in matrix algebra. Each number in a matrix is referred to as an element, and matrices are typically written within brackets.
Determinant Formula
The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated using the formula: \[ \text{det} = ad - bc \]. This formula is derived from the elements of the matrix, where 'a', 'b', 'c', and 'd' represent the individual numbers within the matrix. Hence, knowing where each element is located (i.e., which is 'a', 'b', 'c', and 'd') in the matrix is essential.
Matrix Algebra
Matrix algebra involves operations on matrices, such as addition, subtraction, and multiplication. In the context of determinants, we focus on identifying the individual elements and applying the related formula. Matrix algebra is widely used in various fields such as computer graphics, engineering, and statistics. The determinant specifically helps in understanding the properties of the matrix, such as whether it is invertible (a determinant of zero means the matrix is not invertible).
Precalculus
Precalculus covers a variety of mathematical concepts that prepare students for calculus. This includes understanding matrices and determinants, which are foundational topics. Learning these concepts helps build critical thinking and problem-solving skills. When dealing with the determinant of a 2x2 matrix, the practice of breaking down the problem step-by-step is an excellent way to reinforce mathematical techniques learned in precalculus.

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Most popular questions from this chapter

For the system below, match each determinant in (a)-(d) with its equivalent from choices \(\mathrm{A}-\mathrm{D}\). $$ \begin{aligned} 4 x+3 y-2 z &=1 \\ 7 x-4 y+3 z &=2 \\ -2 x+y-8 z &=0 \end{aligned} $$ \(\begin{array}{lll}\text { (a) } D & \text { (b) } D_{x}\end{array}\) (c) \(D_{y}\) (d) \(D_{z}\) A. \(\left|\begin{array}{rrr}1 & 3 & -2 \\ 2 & -4 & 3 \\ 0 & 1 & -8\end{array}\right|\) B. \(\left|\begin{array}{rrr}4 & 3 & 1 \\ 7 & -4 & 2 \\\ -2 & 1 & 0\end{array}\right|\)C. \(\left|\begin{array}{rrr}4 & 1 & -2 \\ 7 & 2 & 3 \\ -2 & 0 & -8\end{array}\right|\) D. \(\left|\begin{array}{rrr}4 & 3 & -2 \\\ 7 & -4 & 3 \\ -2 & 1 & -8\end{array}\right|\)

Solve each problem. A sparkling-water distributor wants to make up 300 gal of sparkling water to sell for \(\$ 6.00\) per gallon. She wishes to mix three grades of water selling for \(\$ 9.00, \$ 3.00,\) and \(\$ 4.50\) per gallon, respectively. She must use twice as much of the S4.50 water as of the \(\$ 3.00\) water. How many gallons of each should she use?

Solve each problem. A glue company needs to make some glue that it can sell for \(120\) per barrel. It wants to use 150 barrels of glue worth \(100\) per barrel, along with some glue worth \(150\) per barrel and some glue worth \(190\) per barrel. It must use the same number of barrels of \(150\) and \(190\) glue. How much of the \(150\) and \(190\) glue will be needed? How many barrels of \(120\) glue will be produced?

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} &3 x+2 y=-4\\\ &2 x-y=-5 \end{aligned}$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{array}{r} 2 x-y+4 z=-2 \\ 3 x+2 y-z=-3 \\ x+4 y+2 z=17 \end{array}$$
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