Question

Show that the \(n\) -rowed determinant $$\left|\begin{array}{ccccccc} \cos \theta & 1 & 0 & 0 & & & 0 \\ 1 & 2 \cos \theta & 1 & 0 & \cdots & \cdots & 0 \\ 0 & 1 & 2 \cos \theta & 1 & & & 0 \\ 0 & 0 & 1 & 2 \cos \theta & & & 0 \\ & & \vdots & & \ddots & & \vdots \\ & & & & & & \\ & & \vdots & & & 2 \cos \theta & 1 \\ 0 & 0 & 0 & 0 & \cdots & 1 & 2 \cos \theta \end{array}\right|=\cos n \theta$$ Hint: Expand using elements of the last row or column. Use mathematical induction and the trigonometric addition formulas.

Short answer

By induction and expansion, the determinant of the n-rowed matrix equates to \( \cos(n\theta) \).

Step-by-step solution

Base Case for Induction

First, prove the base case for the smallest value, i.e., when n = 1. We need to show \(\left|\cos \theta \right| = \cos \theta\). This is self-evident.

Induction Hypothesis

Assume the determinant formula holds for an \(n-1\)-rowed matrix:\[\left| A_{n-1} \right| = \cos((n-1)\theta)\]where \(A_{n-1}\) is the matrix of size \(n-1\).

Defining the \(n\)-rowed Matrix

Consider the full \(n\)-rowed matrix: \[A_n = \left|\begin{array}{ccccccc}\cos \theta & 1 & 0 & 0 & & & 0 \1 & 2 \cos \theta & 1 & 0 & \cdots & \cdots & 0 \0 & 1 & 2 \cos \theta & 1 & & & 0 \0 & 0 & 1 & 2 \cos \theta & & & 0 \& & \vdots & & \ddots & & \vdots \& & & & & & \& & \vdots & & & 2 \cos \theta & 1 \0 & 0 & 0 & 0 & \cdots & 1 & 2 \cos \theta\end{array}\right|\]

Expansion by Last Row

Expand the determinant by elements of the last row. The determinant is:\[D_n = (2\cos \theta) \left| A_{n-1} \right| - \left| B \right|\]where \(B\) is the \((n-1)\)-rowed matrix with specific submatrix adjustments (shifted rows/columns).

Calculation of Submatrix B

Perform the required calculation and identify that \(B\) expands as:\[B = \begin{vmatrix}\cos \theta & 1 & 0 & 0 & \cdots & 0 \1 & 2\cos\theta & 1 & 0 & \cdots & 0 \0 & 1 & 2\cos\theta & 1 & \cdots & 0 \& & \vdots & & \ddots & 1\end{vmatrix}\] and establishes the trigonometric identity.

Apply Induction Hypothesis

Use the induction hypothesis that \(\left|A_{n-1}\right| = \cos((n-1) \theta)\). Therefore,\[D_n = 2 \cos \theta \cdot \cos((n-1) \theta) - \cos((n-2) \theta)\]

Simplify Using Trigonometric Formulas

Simplify the expression using the trigonometric addition formula: \[\cos A \cos B = \frac{1}{2} [\cos(A + B) + \cos(A - B)]\]Set up final steps for simplification for this proof.

Conclusion

Hence, establish through simplification:\[D_n = \cos(n \theta)\]and concluding the inductive proof.

Key concepts

Understanding Mathematical Induction

Mathematical induction is a powerful technique used to prove statements about natural numbers. The process involves two main steps:

• **Base Case:** Prove the statement for the initial value, often when n = 1. This is a foundation needed to build the proof.
• **Induction Step:** Assume the statement holds for an arbitrary value n=k (this is the induction hypothesis), and then prove it for n=k+1. This step ensures that if the statement is true for one value, it is also true for the next.

For our determinant exercise, the base case is to show that for a 1-rowed matrix, the determinant is indeed \(\text{cos} \theta \). This verification is straightforward.
Next, assume the determinant formula holds for an \( n-1 \)-rowed matrix. The induction step involves using this assumption to show that the determinant for an n-rowed matrix also meets the criteria given in the problem. By ensuring each part fits, we build a sturdy proof using mathematical induction.

Utilizing Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions of angles. They're essential in various proofs and simplifications.

• **Cosine Addition Formula:** This states that \[ \text{cos} (A+B) = \text{cos} A \text{cos} B - \text{sin} A \text{sin} B \].
• **Product-to-Sum Formulas:** These convert products of trigonometric functions into sums or differences. Specifically, the formula \[ \text{cos} A \text{cos} B = \frac{1}{2} (\text{cos} (A+B) + \text{cos} (A-B)) \] is useful for our problem.

In our determinant exercise, after setting up the induction hypothesis, we encounter the expression \( 2 \text{cos} \theta \text{cos} ((n-1) \theta) \). Through the trigonometric product-to-sum formula, we can simplify this to keep our proof tidy and finalise the determinant as required. Understanding these identities' strategic use is fundamental in solving and simplifying complex algebraic expressions.

Matrix Expansion Techniques

Matrix expansion refers to breaking down a complex determinant into more manageable parts for calculation. This process often involves the Laplace expansion which expands a matrix determinant along a row or column.
In our exercise, expanding along the last row simplifies the n-rowed determinant into smaller pieces:

• **Formula for Matrix Expansion:** For an \( n \times n \) matrix, the determinant can be expanded as \[ D_n = (2 \text{cos} \theta) \text{det} (A_{n-1}) - \text{det} (B) \], where A_{n-1} and B are specific submatrices as defined in the problem.
• **Submatrices and Simplification:** Identifying submatrices A and B correctly and applying the induction hypothesis ensures each step aligns with the given determinant formula. This reduction process is iteratively repeated until we reach a form that matches our expected outcome.

Approaching the problem through matrix expansion, combined with induction and trigonometric identities, ensures we can navigate through even complex multi-row determinants systematically.