Problem 46 - Precalculus: Mathematical for Calculus

Question

Find the inverse of the matrix. For what value(s) of \(x\) if any, does the matrix have no inverse? $$\left[\begin{array}{ll} \sec x & \tan x \\ \tan x & \sec x \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is non-invertible for \(x = (2n+1)\pi/2\). Otherwise, the inverse is \( \begin{bmatrix} \sec x & -\tan x \\ -\tan x & \sec x \end{bmatrix} \).

Step 1: Calculate the Determinant

The first step in finding the inverse of a matrix is to calculate its determinant. The formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is \( ad - bc \). For the given matrix, identify \(a = \sec x\), \(b = \tan x\), \(c = \tan x\), and \(d = \sec x\). The determinant is computed as follows: \( \det(A) = (\sec x)(\sec x) - (\tan x)(\tan x) = \sec^2 x - \tan^2 x \).

Step 2: Simplify the Determinant

The identity \( \sec^2 x - \tan^2 x = 1 \) simplifies the determinant to 1 for all \( x \) except where \(\sec x\) is undefined. Thus, the determinant is 1 wherever \( \sec x \) is defined.

Step 3: Identify Non-Invertible Condition

A matrix is non-invertible when its determinant is 0. From Step 2, \( \sec^2 x - \tan^2 x = 1 \) never equals zero, indicating the matrix is not non-invertible for any \( x \) where \( \sec x \) is defined. However, \( \sec x \) is undefined where \( \cos x = 0 \), i.e., \( x = (2n+1)\frac{\pi}{2} \) for any integer \( n \).

Step 4: Conclusion on Inverse

Since the determinant does not become zero, the matrix has an inverse for all \( x \) except where \( \sec x \) is undefined. Therefore, the matrix has no inverse where \( x = (2n+1)\frac{\pi}{2} \).

Step 5: Inverse Matrix Formula

For a 2x2 matrix, if the determinant is non-zero, the inverse is given by: \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Substituting the given matrix values, the inverse is: \( \begin{bmatrix} \sec x & -\tan x \ -\tan x & \sec x \end{bmatrix} \). Having checked the determinant is 1 where \(\sec x \) is defined, divide by 1 to find the inverse.

Key Concepts

Determinant Calculation for Matrix Inverse

The concept of the determinant is fundamental when it comes to finding the inverse of a matrix. For a 2x2 matrix, like the one we have \[ \begin{bmatrix} \sec x & \tan x \ \tan x & \sec x \end{bmatrix} \]. The formula to calculate the determinant is straightforward: \[ \det(A) = ad - bc \]. Here, it's crucial to first identify the elements: \(a = \sec x\), \(b = \tan x\), \(c = \tan x\), and \(d = \sec x\).
Once identified, the determinant calculation proceeds as follows: \(\det(A) = (\sec x)(\sec x) - (\tan x)(\tan x) = \sec^2 x - \tan^2 x\). The key thing to notice here is the resulting expression, \(\sec^2 x - \tan^2 x\), which might seem complex initially but simplifies graciously with a trigonometric identity.
In essence, knowing how to calculate the determinant correctly helps determine whether a matrix is invertible.

Use of Trigonometric Identities

Trigonometric identities are tools that simplify complex expressions into more manageable forms. For this matrix problem, we employ the trigonometric identity \(\sec^2 x - \tan^2 x = 1 \). This identity is derived from the Pythagorean identity, which states \( \tan^2 x + 1 = \sec^2 x \). When rearranged, you later subtract \(\tan^2 x\) from both sides to get the needed identity.
Such identities are particularly handy as they simplify the determinant from \(\sec^2 x - \tan^2 x\) to just 1. This indicates the matrix can have an inverse almost always, given that the determinant won't equal zero unless \(\sec x\) is undefined. Understanding these identities frees one from unnecessary calculation paths and aids in fast problem-solving. Therefore, knowing your trigonometric identities helps not just in algebra but in solving matrix-related problems too.

Matrix Non-Invertibility Conditions

A matrix is considered non-invertible, or singular, when its determinant equals zero. This is a crucial aspect when it comes to matrix inversion, as a non-zero determinant guarantees the existence of an inverse. For the current matrix, although its simplified determinant is 1, signaling no usual case of being zero, we must consider where \(\sec x\) is undefined.
The secant function, \(\sec x\), is undefined wherever \(\cos x = 0\). From our understanding of trigonometry, this occurs when the angle \(x\) is of the form \((2n+1)\frac{\pi}{2}\), where \(n\) is an integer. At these points, \(\sec x\) becomes meaningless, therefore rendering the matrix without an inverse.
Recognizing non-invertibility conditions helps avert mistakes in computation and ensures we are aware of scenarios where the inverse cannot exist. This forms an integral part of matrix theory and applications.