Problem 86
Question
In Exercises 85-90, evaluate the determinant in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. \(\left| \begin{array}{c} 3x^2 & -3y^2 \\ 1 & 1 \end{array} \right|\)
Step-by-Step Solution
Verified Answer
The determinant of the given matrix is \(3x^2 + 3y^2\).
1Step 1: Recognize the format
The task involves a determinant of a 2x2 matrix, which can be calculated using the formula \((ad - bc)\), where a, b, c and d represent the elements of the matrix. Here, a = 3x^2, b = -3y^2, c = 1, and d = 1.
2Step 2: Evaluate the Determinant
The determinant of the matrix is equal to \((a*d - b*c)\). From our above analysis, we can substitute the respective values into the formula. So, determinant = \((3x^2*1) - (-3y^2*1)\).
3Step 3: Simplify the Expression
The previous step can be simplified further to: determinant = \(3x^2 + 3y^2\).
Key Concepts
Determinant of a Matrix2x2 MatrixMatrix Operations in CalculusVariable Substitution in Calculus
Determinant of a Matrix
The determinant of a matrix is a special number that provides a lot of information about the matrix itself. For instance, it can tell us whether the matrix is invertible, i.e., if an inverse matrix exists, and it can also indicate the volume scaling factor when the matrix is considered a transformation.
When evaluating the determinant for a 2x2 matrix, the process is fairly straightforward. The formula is: \(ad - bc\), where the letters represent the entries of the matrix in the order: \(a, b\) in the first row, and \(c, d\) in the second row. So, for the given matrix \(\left| \begin{array}{cc} 3x^2 & -3y^2 \ 1 & 1 \end{array} \right|\), the determinant is calculated by multiplying the entries from the top left to the bottom right (denoted as 'ad'), and subtracting the product of the entries from the top right to the bottom left (denoted as 'bc').
To assist students who might face difficulties in understanding the determinant concept, it is essential to remember that it's more than a mere number—it reflects the geometrical properties of the matrix and has numerous applications, especially in calculus where it's used for variable transformation.
When evaluating the determinant for a 2x2 matrix, the process is fairly straightforward. The formula is: \(ad - bc\), where the letters represent the entries of the matrix in the order: \(a, b\) in the first row, and \(c, d\) in the second row. So, for the given matrix \(\left| \begin{array}{cc} 3x^2 & -3y^2 \ 1 & 1 \end{array} \right|\), the determinant is calculated by multiplying the entries from the top left to the bottom right (denoted as 'ad'), and subtracting the product of the entries from the top right to the bottom left (denoted as 'bc').
To assist students who might face difficulties in understanding the determinant concept, it is essential to remember that it's more than a mere number—it reflects the geometrical properties of the matrix and has numerous applications, especially in calculus where it's used for variable transformation.
2x2 Matrix
A 2x2 matrix is one of the simplest forms of matrices, consisting of 2 rows and 2 columns. It is a rectangular array of numbers, and its properties and operations are foundations for higher level mathematics. Evaluating the determinant of a 2x2 matrix is an essential skill necessary for students to proceed to more complex matrix operations.
For learning purposes, it is beneficial to visualize the 2x2 matrix and the determinant calculation as a cross-multiplication where the diagonals are multiplied and one product is subtracted from the other. This operation, although simple for a 2x2 matrix, sets the groundwork for understanding larger matrices where the process of finding a determinant is significantly more complex.
For learning purposes, it is beneficial to visualize the 2x2 matrix and the determinant calculation as a cross-multiplication where the diagonals are multiplied and one product is subtracted from the other. This operation, although simple for a 2x2 matrix, sets the groundwork for understanding larger matrices where the process of finding a determinant is significantly more complex.
Matrix Operations in Calculus
In the realm of calculus, matrices are not just static arrays of numbers but are dynamic tools for representing and solving various problems. Matrix operations, such as addition, subtraction, multiplication, and finding determinants, are important in linear transformations, systems of equations, and variable changes in integrations.
In the context of calculus, matrices can often represent systems of partial derivatives, known as Jacobian matrices, which are crucial when performing variable substitution in multiple integrals. Understanding how to compute and work with determinants in this setting allows students to transition smoothly between coordinate systems and apply the correct scaling factors for integration.
In the context of calculus, matrices can often represent systems of partial derivatives, known as Jacobian matrices, which are crucial when performing variable substitution in multiple integrals. Understanding how to compute and work with determinants in this setting allows students to transition smoothly between coordinate systems and apply the correct scaling factors for integration.
Variable Substitution in Calculus
Variable substitution, also known as u-substitution, is a technique used in calculus to simplify the integration process by changing complex expressions into simpler ones. When the change involves more than one variable, as in multiple integrals, the Jacobian determinant comes into play.
This determinant helps to adjust the area or volume elements during the substitution. It factors in how an infinitesimal region of space changes shape and size under the transformation. Thus, for students to proficiently perform these substitutions, they would need to have a solid grasp of evaluating determinants, especially in the context of functional entries as seen in our exercise example.
This determinant helps to adjust the area or volume elements during the substitution. It factors in how an infinitesimal region of space changes shape and size under the transformation. Thus, for students to proficiently perform these substitutions, they would need to have a solid grasp of evaluating determinants, especially in the context of functional entries as seen in our exercise example.
Other exercises in this chapter
Problem 85
THINK ABOUT IT In Exercises 79-86, let matrices \(A\), \(B\), \(C\), and \(D\), be of orders and \(2 \times 3\), \(2 \times 3\), \(3 \times 2\) and \(2 \times 2
View solution Problem 85
In Exercises 85-90, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the sys
View solution Problem 86
THINK ABOUT IT In Exercises 79-86, let matrices \(A\), \(B\), \(C\), and \(D\), be of orders and \(2 \times 3\), \(2 \times 3\), \(3 \times 2\) and \(2 \times 2
View solution Problem 86
In Exercises 85-90, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the sys
View solution