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Chapter 9: Problem 57

Find the determinant of the matrix \(\left[\begin{array}{rr}3 & -2 \\ -1 & 4\end{array}\right]\).

Short Answer

Expert verified
The determinant is 10.

Step by step solution

01

- Identify the Matrix Elements

Given the matrix \(\begin{bmatrix} 3 & -2 \ -1 & 4 \end{bmatrix}\), identify the elements: \(a = 3, b = -2, c = -1, d = 4\).
02

- Recall the Determinant Formula for a 2x2 Matrix

The formula to find the determinant for a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by \( \text{det}(A) = ad - bc \).
03

- Substitute the Elements into the Formula

Substitute \(a = 3\), \(b = -2\), \(c = -1\), and \(d = 4\) into the determinant formula: \[ \text{det}(A) = 3 \times 4 - (-2) \times (-1) \]
04

- Perform the Arithmetic Operations

Calculate the products: \( 3 \times 4 = 12\) and \( (-2) \times (-1) = 2 \). Thus, \[ \text{det}(A) = 12 - 2 \]
05

- Find the Determinant

Subtract the two results: \(\text{det}(A) = 12 - 2 = 10\).

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Key Concepts

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

2x2 Matrix
Let's start by understanding what a 2x2 matrix is. A matrix is a rectangular array of numbers arranged in rows and columns. A 2x2 matrix specifically has exactly two rows and two columns.

It can be visually represented as:

\[ \begin{bmatrix} a & b \ c & d \ \ \end{bmatrix} \]

Here, the elements in the first row are 'a' and 'b', and the elements in the second row are 'c' and 'd'. Matrices are essential tools in various fields, including mathematics, physics, and engineering.

In our given exercise, we have the matrix:

\[ \begin{bmatrix} 3 & -2 \ -1 & 4 \end{bmatrix} \]

This means:
  • 'a' is 3
  • 'b' is -2
  • 'c' is -1
  • 'd' is 4
Matrix Determinant Formula
To find the determinant of a 2x2 matrix, we use a specific formula. The determinant helps us understand properties like invertibility and is a scalar value.

For any 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula:

\[ \text{det}(A) = ad - bc \]

The product of the elements on the main diagonal (top-left to bottom-right) is subtracted by the product of the elements on the other diagonal.

For our given matrix, substitute the values:

'a' is 3, 'b' is -2, 'c' is -1, and 'd' is 4.

This gives us:

\[ \text{det}(A) = 3 \times 4 - (-2) \times (-1) \]
Matrix Arithmetic Operations
Understanding matrix arithmetic operations is crucial as these operations often appear in linear algebra.

For our problem, let's focus on the multiplication and subtraction operations:

  • Multiplying 3 and 4 gives us 12 as:
    \[ 3 \times 4 = 12 \]
  • Multiplying -2 and -1 yields 2 (since a negative times a negative is positive):
    \[ (-2) \times (-1) = 2 \]
  • Finally, to find the determinant, subtracting these results gives us:

    \[12 - 2 = 10 \]
Performing these arithmetic steps correctly will ensure you arrive at the correct determinant.
Linear Algebra
Linear algebra is the branch of mathematics that deals with vector spaces and linear mappings between them. It includes the study of lines, planes, and subspaces, but also more complex structures.

Determinants play a significant role in linear algebra. They help determine:
  • If a matrix is invertible (non-zero determinant indicates an invertible matrix)
  • Solutions to systems of linear equations
  • Properties related to linear transformations
In our example, finding the determinant of the given 2x2 matrix is part of understanding its behavior in the context of linear transformations and systems of equations.

The determinant value (10 in our case) signals that the matrix is invertible and has unique solutions in the corresponding system of linear equations.

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Most popular questions from this chapter

Solve each problem, using two linear equations in two variables and Cramer's rule. Average Salary The average salary for the president and vice-president of Intermax Office Supply is $$ 200,000\(. If the president's salary is $$ 100,000\) more than the vice president's, then what is the salary of each?

Find each product if possible. $$\left[\begin{array}{rr}\frac{1}{4} & \frac{1}{2} \\\\\frac{1}{8} & -\frac{1}{2}\end{array}\right]\left[\begin{array}{rr} -\frac{1}{2} & \frac{1}{4} \\\\\frac{1}{2} & \frac{1}{4}\end{array}\right]$$

Write a matrix equation of the form \(A X=B\) that corresponds to each system of equations. $$\begin{aligned}&x-y=-7\\\&x+2 y=8\end{aligned}$$

Find each product if possible. $$\left[\begin{array}{ll}\frac{1}{2} & \frac{1}{3} \\\\\frac{1}{4} & \frac{1}{5}\end{array}\right]\left[\begin{array}{ll}-8 & 12 \\ -5 & 15\end{array}\right]$$

Solve each problem, using Cramer's rule and a graphing calculator. Gasoline Sales The Runway Deli sells regular unleaded, plus unleaded, and supreme unleaded Shell gasoline. The number of gallons of each grade and the total receipts for gasoline are shown in the table for the first three weeks of February. What was the price per gallon for each grade? $$\begin{array}{|c|c|c|c|c|} \hline & \text { Regular } & \text { Plus } & \text { Supreme } & \text { Receipts } \\ \hline \text { Week 1 } & 1270 & 980 & 890 & \$ 12,204.86 \\ \text { Week 2 } & 1450 & 1280 & 1050 & \$ 14,698.22 \\ \text { Week 3 } & 1340 & 1190 & 1060 & \$ 13,969.41 \\ \hline \end{array}$$
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