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Chapter 7: Problem 14

For the following exercises, find the determinant. \(\begin{array}{rr}-1.1 & 0.6 \\ 7.2 & -0.5\end{array} \mid\)

Short Answer

Expert verified
The determinant of the matrix is -3.77.

Step by step solution

01

Identify the Matrix Elements

The matrix given is a 2x2 matrix: \[\begin{bmatrix}a & b \c & d \end{bmatrix} = \begin{bmatrix}-1.1 & 0.6 \7.2 & -0.5 \end{bmatrix}\]Here, \(a = -1.1\), \(b = 0.6\), \(c = 7.2\), and \(d = -0.5\).
02

Determine the Formula for Determinant of a 2x2 Matrix

The determinant \(D\) of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by the formula:\[D = ad - bc\]
03

Calculate Each Component

Using the formula, we need to calculate each part:- \(ad\) is calculated as: \(-1.1 \times -0.5 = 0.55\)- \(bc\) is calculated as: \(0.6 \times 7.2 = 4.32\)
04

Compute the Determinant

Substitute the values of \(ad\) and \(bc\) into the determinant formula:\[D = ad - bc = 0.55 - 4.32\]Calculate the result:\[D = -3.77\]

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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 foundational concept in linear algebra. It is a grid made up of 2 rows and 2 columns, forming a square matrix. Each element in this grid is a number, and these numbers can represent various types of data, such as coefficients in a set of linear equations or transformations in geometry. The structure of a 2x2 matrix is commonly expressed as: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\] Here, \(a\), \(b\), \(c\), and \(d\) are the elements that make up the matrix. In practical exercises, these elements will often be given specific numerical values, as seen in problems involving the computation of determinants.
Matrix Elements
Matrix elements are the individual numbers contained within the matrix. In a 2x2 matrix, these elements are crucial for performing operations such as finding the determinant. Each position in the matrix is labeled by its row and column, which helps identify and differentiate between the elements:
  • \(a\) is the element in the first row, first column.
  • \(b\) is in the first row, second column.
  • \(c\) is in the second row, first column.
  • \(d\) is in the second row, second column.
Knowing the position of these elements makes it easier to apply various matrix operations and ensures that the calculations are accurate.
Determinant Formula
Every 2x2 matrix has a determinant, which is a special number that provides important information about the matrix. This number is computed using the determinant formula. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the formula is:\[D = ad - bc\]Where:
  • \(ad\) is the product of the elements on the main diagonal (from top left to bottom right).
  • \(bc\) is the product of the elements on the minor diagonal (from top right to bottom left).
The resulting number, \(D\), helps identify properties of the matrix. For instance, if \(D\) is zero, the matrix is singular, meaning it does not have an inverse.
Computing Determinants
Once the determinant formula is identified, the next step is computing the determinant by substituting in the matrix elements. This step turns the abstract idea of a determinant into a tangible calculation.Here's how it works: you multiply the elements on the main diagonal and subtract the product of the elements on the minor diagonal.For example, with the matrix elements given as \(-1.1, 0.6, 7.2,\) and \(-0.5\), the calculations would be:
  • Calculate \(ad\) as \(-1.1 \times -0.5 = 0.55\).
  • Calculate \(bc\) as \(0.6 \times 7.2 = 4.32\).
  • Determine \(D\) with the formula: \(0.55 - 4.32 = -3.77\).
Thus, the determinant \(D\) is \(-3.77\), providing insight into the matrix's properties, such as its invertibility and the scaling factor it applies in transformations.

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Most popular questions from this chapter

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution. Th ee numbers add up to \(106 .\) The fi st number is 3 less than the second number. The third number is 4 more than the fi st number.

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{aligned} x+2 y-4 z &=-1 \\ 7 x+3 y+5 z &=26 \\ -2 x-6 y+7 z &=-6 \end{aligned} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{aligned} 8 x-2 y &=-3 \\ -4 x+6 y &=4 \end{aligned} $$

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer鈥檚 Rule. At a market, the three most popular vegetables make up 53\(\%\) of vegetable sales. Corn has 4\(\%\) higher sales than broccoli, which has 5\(\%\) more sales than onions. What percentage does each vegetable have in the market share?

For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate- covered cashews. The nutritional information for these items is shown in Table 1 . $$ \begin{array}{|c|c|c|c|} \hline & \text { Fat (g) } & \text { Protein (g) } & \text { Carbohydrates (g) } \\ \hline \text { Almonds (10) } & 6 & 2 & 3 \\ \hline \text { Cranberries (10) } & 0.02 & 0 & 8 \\ \hline \text { Cashews (10) } & 7 & 3.5 & 5.5 \\ \hline \end{array} $$ For the "energy-booster" mix, there are 1,000 pieces in the mix, containing \(145 \mathrm{~g}\) of protein and \(625 \mathrm{~g}\) of carbohydrates. If the number of almonds and cashews summed together is equivalent to the amount of cranberries, how many of each item is in the trail mix?
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