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Chapter 9: Problem 6

Find the value of each determinant. $$\left|\begin{array}{ll} 0 & 2 \\ 1 & 5 \end{array}\right|$$

Short Answer

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Step by step solution

01

Recall the Formula for a 2x2 Determinant

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

Identify the Elements in the Matrix

For the given matrix \(\begin{bmatrix} 0 & 2 \ 1 & 5 \end{bmatrix}\), identify \(a = 0\), \(b = 2\), \(c = 1\), and \(d = 5\).
03

Substitute the Values into the Determinant Formula

Substitute \(a = 0\), \(b = 2\), \(c = 1\), and \(d = 5\) into the formula: \(0 \times 5 - 2 \times 1\).
04

Simplify the Expression

Calculate the expression: \(0 \times 5 = 0\) and \(2 \times 1 = 2\). Thus, \(0 - 2 = -2\).
05

State the Determinant

The determinant of the matrix \(\begin{bmatrix} 0 & 2 \ 1 & 5 \end{bmatrix}\) is \(-2\).

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Key Concepts

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

matrix operations
Matrix operations are fundamental in mathematics and have applications in various fields such as physics, engineering, and computer science. They include addition, subtraction, multiplication, and finding the determinant.
Understanding these operations helps in solving systems of equations, transforming geometric objects, and more.
In this exercise, we focus specifically on the operation of finding the determinant, which is crucial for understanding more complex concepts in linear algebra.
calculating determinants
Calculating determinants is a key operation when dealing with matrices. The determinant of a matrix is a special number that can offer insights into the properties of the matrix.
For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as: \[ ad - bc \].
Steps to calculate the determinant are:
  • Identify the elements in the matrix (a, b, c, d).
  • Substitute these values into the determinant formula.
  • Simplify the expression to find the determinant.

For the matrix \(\begin{bmatrix} 0 & 2 \ 1 & 5 \end{bmatrix}\), we have a=0, b=2, c=1, and d=5.
Applying the formula, we get: \[ 0 \times 5 - 2 \times 1 = -2 \].
Therefore, the determinant for this matrix is -2.
matrices in precalculus
In precalculus, matrices provide a systematic way to handle and solve linear equations. They are used to organize coefficients of variables in a compact form.
Additionally, understanding matrices and their properties, like the determinant, lays the groundwork for more advanced studies in calculus and linear algebra.
By knowing how to calculate determinants, you can determine if a matrix has an inverse, which is crucial for solving equations.
Practice regularly with different types of matrices to become proficient in these essential operations.
These foundational skills are built upon later in higher-level mathematics.

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Most popular questions from this chapter

Solve each problem. In certain parts of the Rocky Mountains, deer provide the main food source for mountain lions. When the deer population is large, the mountain lions thrive. However, a large mountain lion population reduces the size of the deer population. Suppose the fluctuations of the two populations from year to year can be modeled with the matrix equation $$ \left[\begin{array}{c} m_{n+1} \\ d_{n+1} \end{array}\right]=\left[\begin{array}{rr} 0.51 & 0.4 \\ -0.05 & 1.05 \end{array}\right]\left[\begin{array}{l} m_{n} \\ d_{n} \end{array}\right] $$ The numbers in the column matrices give the numbers of animals in the two populations after \(n\) years and \(n+1\) years, where the number of deer is measured in hundreds. (a) Give the equation for \(d_{n+1}\) obtained from the second row of the square matrix. Use this equation to determine the rate at which the deer population will grow from year to year if there are no mountain lions. (b) Suppose we start with a mountain lion population of 2000 and a deer population of \(500,000\) (that is, 5000 hundred deer). How large would each population be after 1 yr? 2 yr? (c) Consider part (b) but change the initial mountain lion population to \(4000 .\) Show that the populations would both grow at a steady annual rate of 1.01

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

Concept Check Write a system of inequalities for which the graph is the region in the first quadrant inside and including the circle with radius 2 centered at the origin, and above (not including) the line that passes through the points \((0,-1)\) and \((2,2)\)

Solve each system by using the inverse of the coefficient matrix. $$\begin{aligned} &2 x-3 y=10\\\ &2 x+2 y=5 \end{aligned}$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} -2 x-2 y+3 z &=4 \\ 5 x+7 y-z &=2 \\ 2 x+2 y-3 z &=-4 \end{aligned}$$
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