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Chapter 6: Problem 13

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}4 & -7 & 8 \\\2 & 1 & 3 \\\\-6 & 3 & 0\end{array}\right]$$

Short Answer

Expert verified
The determinant is 186.

Step by step solution

01

Understand the determinant formula

The determinant of a 3x3 matrix \( A \), denoted as \( \text{det}(A) \), is calculated using the formula: \[ \text{det}(A) = a(ei 鈭� fh) 鈭� b(di 鈭� fg) + c(dh 鈭� eg) \]where the matrix is \[ A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \] and \( a, b, c, d, e, f, g, h, i \) are the elements of the matrix.
02

Identify matrix elements

For the matrix \[ \begin{bmatrix} 4 & -7 & 8 \ 2 & 1 & 3 \ -6 & 3 & 0 \end{bmatrix} \],assign the elements as follows:\( a = 4, b = -7, c = 8, d = 2, e = 1, f = 3, g = -6, h = 3, \) and \( i = 0 \).
03

Apply the determinant formula

Substitute the values into the determinant formula:\[\text{det}(A) = 4(1 imes 0 - 3 imes 3) - (-7)(2 imes 0 - 3 imes -6) + 8(2 imes 3 - 1 imes -6)\]Simplify each term step-by-step.
04

Simplify each term individually

Calculate each term in the expression: 1. Calculate \( 4(1 imes 0 - 3 \times 3) = 4(0 - 9) = 4(-9) = -36 \).2. Calculate \( -(-7)(2 imes 0 - 3 imes -6) = 7(0 + 18) = 7 \times 18 = 126 \).3. Calculate \( 8(2 \times 3 - 1 \times -6) = 8(6 + 6) = 8 \times 12 = 96 \).
05

Combine the results of the terms

Add all the results together to find the determinant:\(\text{det}(A) = -36 + 126 + 96 = 186\).

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Key Concepts

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

3x3 Matrix
Understanding a 3x3 matrix is crucial when learning about determinants and matrix algebra. A 3x3 matrix consists of three rows and three columns of numbers. This is often seen as a square matrix, meaning the number of rows is equal to the number of columns.

The 3x3 matrix can be represented as follows:
  • The first row contains the elements: \( a, b, c \)
  • The second row contains the elements: \( d, e, f \)
  • The third row contains the elements: \( g, h, i \)
These elements are crucial in calculating the determinant, which provides important information about the matrix, such as whether it is invertible. A non-zero determinant indicates an invertible matrix, while a zero determinant reflects a singular (non-invertible) matrix.

Visualizing or drawing the matrix structure can help understand how each element's position contributes to the determinant's calculation, especially when using the formula.
Matrix Algebra
Matrix algebra involves operations that can be performed on matrices, such as addition, subtraction, multiplication, and finding determinants. Calculating the determinant is a fundamental concept in matrix algebra, as it has implications in solving systems of linear equations and understanding matrix properties.

Determinants are calculated using specific formulas that depend on the size of the matrix. For a 3x3 matrix, the formula incorporates products and sums of its elements. This method involves evaluating certain products and their respective permutations:
  • Calculate the products including \( a(ei 鈭� fh) \)
  • Followed by \(- b(di 鈭� fg) \)
  • Lastly, add \(+ c(dh 鈭� eg) \)
In matrix algebra, understanding how to properly manipulate these elements is key. This ensures accurate computations and a better grasp of the matrix's characteristics, such as its ability to transform or solve equations.
Elementary Algebra
Elementary algebra is the foundation upon which more advanced mathematical concepts are built, including matrices and determinants. It involves basic mathematical operations such as addition, subtraction, multiplication, and division.

In the context of determinant calculation, elementary algebra handles the arithmetic part of the formula. It ensures that the numbers are correctly calculated step-by-step:
  • Add or subtract terms such as \( (ei 鈭� fh), (di 鈭� fg), (dh 鈭� eg) \)
  • Multiply these results by \( a, b, c \)
  • Finally, sum up these products to obtain the determinant value.
The step-by-step approach in elementary algebra is vital in minimizing errors, especially in larger matrices, where the number of calculations increases. Emphasizing clarity and precision in calculations bridges the gap between understanding simple arithmetic and mastering more complex matrix operations.

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Most popular questions from this chapter

The break-even point for a company is the point where costs equal revenues. If both cost and revenue are expressed as linear equations, the break-even point is the solution of a linear system. In each exercise, \(C\) represents cost in dollars to produce x items, and R represents revenue in dollars from the sale of \(x\) items. Use the substitution method to find the break-even point in each case-that is, the point where \(C=R .\) Then find the value of \(C\) and \(R\) at that point. $$\begin{aligned}&C=20 x+10,000\\\&R=30 x-11,000\end{aligned}$$

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \((A B) C=A(B C)\) (associative property)

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A line and a circle; two points.

As the price of a product increases, businesses usually increase the quantity manufactured. However, as the price increases, consumer demand-or the quantity of the product purchased by consumers-usually decreases. The price we see in the market place occurs when the quantity supplied and the quantity demanded are equal. This price is called the equilibrium price and this demand is called the equilibrium demand. (Refer to Exercise 92 .) Suppose that supply is related to price by \(p=\frac{1}{10} q\) and that demand is related to price by \(p=15-\frac{2}{3} q,\) where \(p\) is price in dollars and \(q\) is the quantity supplied in units. (a) Determine the price at which 15 units would be supplied. Determine the price at which 15 units would be demanded. (b) Determine the equilibrium price at which the quantity supplied and quantity demanded are equal. What is the demand at this price?

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &e^{-x}-y \leq 1\\\ &x-2 y \geq 4 \end{aligned}$$
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