Chapter 3: Problem 13
Show that $$\left|\begin{array}{ccc} \cos \theta & 1 & 0 \\ 1 & 2 \cos \theta & 1 \\ 0 & 1 & 2 \cos \theta \end{array}\right|=\cos 3 \theta$$
Short Answer
Expert verified
The determinant evaluates to \cos(3 \theta)\.
Step by step solution
01
- Write down the determinant
The given determinant is \[\begin{vmatrix} \begin{array}{ccc} \begin{array}{c} \ \cos \theta & 1 & 0 \ 1 & 2 \cos \theta & 1 \ 0 & 1 & 2 \cos \theta \begin{array}{a} \begin{array}{ccc} \begin{array}{ccc} \end{array}\end{array}\end{array}\end{array} \begin{array}{c} \end{array} \end{array} \end{array} \begin{array}{ccc} \end{array}\begin{array}{ccc} \begin{array}{ccc} \end{array} \begin{array}{ccc} \end{array}\end{vmatrix}\].
02
- Expand the determinant along the first row
Using the formula for the determinant expansion along the first row, we get: \[\begin{vmatrix} \begin{array}{ccc} \cos \theta & 1 & 0 \begin{array}{c} \end{array}\begin{array}{c} \end{array}\end{array}\].
03
- Calculate minors
Calculate each minor that results from removing the first row and one column at a time. This gives: \[ \cos \theta * \begin{vmatrix} 2 \cos \theta & 1 \ 1 & 2 \cos \theta \end{vmatrix} - 1 * \begin{vmatrix} 1 & 1 \ 0 & 2 \cos \theta \end{vmatrix} \].
04
- Evaluate 2x2 determinants
Calculate the smaller 2x2 determinants: \[\begin{vmatrix} 2 \cos \theta & 1 \ 1 & 2 \cos \theta\end{vmatrix} = 4 \cos^2 \theta - 1 \] and \[\begin{vmatrix} 1 & 1 \ 0 & 2 \cos \theta\end{vmatrix} = 2 \cos \theta \].
05
- Substitute back to original determinant
Substituting back the calculated 2x2 determinants, we get: \[\cos \theta (4 \cos^2 \theta - 1) - 2 \cos \theta= 4 \cos^3 \theta - \cos \theta - 2 \cos \theta = 4 \cos^3 \theta - 3 \cos \theta \].
06
- Simplify the expression
Recognize the resulting expression as the triple-angle formula for cosine: \[\cos 3\theta = \cos(3 \theta)\]. Therefore, we have shown that: \[\begin{vmatrix} \cos \theta & 1 & 0 \ 1 & 2 \cos \theta & 1 \ 0 & 1 & 2 \cos \theta \end{vmatrix} = \cos 3 \theta \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
matrix determinant
A matrix determinant is a crucial concept in linear algebra. It is a scalar value that provides important information about a square matrix.
In the provided exercise, we work with a 3x3 matrix, showing the determinant equals \(\text{cos} 3θ\). The determinant helps determine various properties of a matrix, such as whether the matrix is invertible. If the determinant is zero, the matrix cannot be inverted.
Determinants can be calculated using various methods, such as expanding along a row or a column, as shown in the solution. For larger matrices, this involves calculating smaller determinants until you reach 2x2 matrices.
The formula for a 2x2 matrix determinant is simple: \(\begin{vmatrix} a & b \ c & d \ \end{vmatrix} = ad - bc\). By building upon this, one can use cofactor expansion to calculate the determinant of larger matrices readily.
In the provided exercise, we work with a 3x3 matrix, showing the determinant equals \(\text{cos} 3θ\). The determinant helps determine various properties of a matrix, such as whether the matrix is invertible. If the determinant is zero, the matrix cannot be inverted.
Determinants can be calculated using various methods, such as expanding along a row or a column, as shown in the solution. For larger matrices, this involves calculating smaller determinants until you reach 2x2 matrices.
The formula for a 2x2 matrix determinant is simple: \(\begin{vmatrix} a & b \ c & d \ \end{vmatrix} = ad - bc\). By building upon this, one can use cofactor expansion to calculate the determinant of larger matrices readily.
trigonometric identities
Trigonometric identities, such as the triple-angle identity, are essential in simplifying and solving trigonometric equations.
In the solution, we identified that \(\text{cos} 3θ\) is the simplified form of the determinant expression. This identity is one of several that link various trigonometric functions across different angles. Common identities include the Pythagorean identity, double-angle formulas, and sum and difference formulas.
The triple-angle formula for cosine is \(\text{cos} 3θ = 4 \text{cos}^3θ - 3 \text{cos}θ\). Recognizing and using such identities is instrumental in both pure and applied mathematics, from simplifying algebraic expressions to solving complex physics problems.
Trigonometric identities often provide ways to rewrite and simplify expressions, making it easier to find solutions to trigonometric equations.
In the solution, we identified that \(\text{cos} 3θ\) is the simplified form of the determinant expression. This identity is one of several that link various trigonometric functions across different angles. Common identities include the Pythagorean identity, double-angle formulas, and sum and difference formulas.
The triple-angle formula for cosine is \(\text{cos} 3θ = 4 \text{cos}^3θ - 3 \text{cos}θ\). Recognizing and using such identities is instrumental in both pure and applied mathematics, from simplifying algebraic expressions to solving complex physics problems.
Trigonometric identities often provide ways to rewrite and simplify expressions, making it easier to find solutions to trigonometric equations.
linear algebra
Linear Algebra is a branch of mathematics focused on vectors, vector spaces, and linear transformations.
Determinants and matrices are core components of linear algebra. They help describe systems of linear equations, transformations, and the behavior of vector spaces. Understanding how to compute determinants and apply them to linear transformations provides insight into the structure and solutions of linear systems.
Linear algebra also plays a vital role in various applications like computer graphics, engineering, physics, and more. It is foundational to understanding advanced topics such as eigenvalues, eigenvectors, and diagonalization.
The solution to the exercise highlights practical aspects of linear algebra, showing how expanding determinants and simplifying trigonometric expressions can solve complex problems. Mastering these concepts provides a solid base for diving deeper into the world of mathematics and its applications.
Determinants and matrices are core components of linear algebra. They help describe systems of linear equations, transformations, and the behavior of vector spaces. Understanding how to compute determinants and apply them to linear transformations provides insight into the structure and solutions of linear systems.
Linear algebra also plays a vital role in various applications like computer graphics, engineering, physics, and more. It is foundational to understanding advanced topics such as eigenvalues, eigenvectors, and diagonalization.
The solution to the exercise highlights practical aspects of linear algebra, showing how expanding determinants and simplifying trigonometric expressions can solve complex problems. Mastering these concepts provides a solid base for diving deeper into the world of mathematics and its applications.