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Chapter 5: Problem 8

Evaluate the given determinants. $$\left|\begin{array}{rr} -4 & 7 \\ 1 & -3 \end{array}\right|$$

Short Answer

Expert verified
The determinant of the matrix is 5.

Step by step solution

01

Understand the Form of the Determinant

The determinant is a 2x2 matrix given by:\[\left|\begin{array}{rr} -4 & 7 \ 1 & -3 \end{array}\right|\] where the elements are arranged in two rows and two columns.
02

Apply the Formula for a 2x2 Determinant

For a 2x2 matrix determinant \( \left|\begin{array}{cc} a & b \ c & d \end{array}\right| \), the determinant is calculated as \( ad - bc \).
03

Substitute the Matrix Elements into the Formula

In the given matrix, let \( a = -4 \), \( b = 7 \), \( c = 1 \), and \( d = -3 \). Substitute these into the formula:\[(-4)(-3) - (7)(1)\]
04

Perform the Calculations

Calculate the products within the formula:1. \((-4) \times (-3) = 12\)2. \(7 \times 1 = 7\)Subtract the second product from the first:\[12 - 7 = 5\]
05

Conclude the Result

The determinant of the given matrix is 5. Thus, the evaluated determinant is 5.

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Key Concepts

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

Understanding a 2x2 Matrix
A 2x2 matrix is a simple form of a matrix used in many mathematical computations and real-world applications. It is called "2x2" because it consists of two rows and two columns. You can visualize it like a small table with four boxes.
Each element in these boxes is often represented by a letter, like \(a\), \(b\), \(c\), and \(d\). In our given problem, the matrix is:
  • -4 in the first row, first column
  • 7 in the first row, second column
  • 1 in the second row, first column
  • -3 in the second row, second column
This arrangement makes 2x2 matrices easy to handle, providing a perfect starting point for learning matrix operations.
The Mathematical Formula for a Determinant
A determinant is a special number calculated from a square matrix. For a 2x2 matrix, there is a straightforward formula to find the determinant. Using the general form \(\begin{bmatrix} a & b \ c & d \\end{bmatrix}\), the formula to find the determinant is given by:
  • \(ad - bc\)
This formula involves two steps:
  1. Multiply the elements of the main diagonal (top-left to bottom-right) together: \(a \times d\).
  2. Multiply the elements of the other diagonal (top-right to bottom-left) together: \(b \times c\).
Finally, subtract the second result from the first. This method provides a quick way to calculate and assess a determinant's value, which in turn helps solve various mathematical problems involving matrices.
Performing Matrix Operations to Find Determinants
Matrix operations involve actions such as addition, subtraction, multiplication, and finding determinants. For our specific task of finding the determinant of a 2x2 matrix, it's crucial to follow a few calculated operations. For the matrix \[ \begin{bmatrix} -4 & 7 \ 1 & -3 \end{bmatrix} \], we use the determinant formula \(ad - bc\) derived previously.
  • Start by multiplying \(-4\) (top left) with \(-3\) (bottom right), to get: \((-4)(-3) = 12\)
  • Next, multiply \(7\) (top right) with \(1\) (bottom left), to get: \(7 \times 1 = 7\)
  • Finally, subtract the second product from the first: \(12 - 7 = 5\)
The result, 5, is the determinant of the matrix. Efficiently performing these operations can simplify complex calculations and reveal important matrix characteristics, often used in fields such as physics and engineering.

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Most popular questions from this chapter

Solve the given problems by determinants. In a laboratory experiment to measure the acceleration of an object, the distances traveled by the object were recorded for three different time intervals. These data led to the following equations: $$\begin{array}{l} s_{0}+2 v_{0}+2 a=20 \\ s_{0}+4 v_{0}+8 a=54 \\ s_{0}+6 v_{0}+18 a=104 \end{array}$$ Here, \(s_{0}\) is the initial displacement (in \(\mathrm{ft}\) ), \(v_{0}\) is the initial velocity (in \(\mathrm{ft} / \mathrm{s})\), and \(a\) is the acceleration (in \(\mathrm{ft} / \mathrm{s}^{2}\) ). Find \(s_{0}, v_{0},\) and \(a\).

In order to make the coefficients easier to work with, first multiply each term of the equation or divide each term of the equation by a number selected by inspection. Then proceed with the solution of the system by an appropriate algebraic method. $$\begin{array}{l} 250 R+225 Z=400 \\ 375 R-675 Z=325 \end{array}$$

Show that the given systems of equations have either an unlimited number of solutions or no solution. If there is an unlimited number of solutions, find one of them. $$\begin{aligned}&3 x+3 y-2 z=2\\\&2 x-y+z=1\\\&x-5 y+4 z=-3\end{aligned}$$

Solve the given systems of equations by either method of this section. $$\begin{aligned} &2 x-y=5\\\ &6 x+2 y=-5 \end{aligned}$$

Solve the given systems of equations by the method of elimination by substitution. $$\begin{aligned} &y=2 x+10\\\ &2 x+y=-2 \end{aligned}$$
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