Problem 44 - 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? $$\left[\begin{array}{cc} x & 1 \\ -x & \frac{1}{x-1} \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix has no inverse when \(x = 0\) or \(x = 1\).

Step 1: Understanding the Condition for Inverse

For a 2x2 matrix \(\left[\begin{array}{cc} a & b \ c & d \end{array}\right] \) to have an inverse, the determinant \(ad - bc\) must not be zero. If the determinant is zero, the matrix does not have an inverse.

Step 2: Calculate the Determinant

Let's calculate the determinant of the matrix \(\begin{bmatrix} x & 1 \ -x & \frac{1}{x-1} \end{bmatrix} \). The determinant is: \[ \det = x \cdot \frac{1}{x-1} - (1)(-x) = \frac{x}{x-1} + x \].

Step 3: Simplifying the Determinant Expression

To simplify \( \frac{x}{x-1} + x \), express \( x \) with a common denominator:\[ \frac{x}{x-1} + \frac{x(x-1)}{x-1} = \frac{x + x^2 - x}{x-1} = \frac{x^2}{x-1} \].

Step 4: Find the Value(s) where the Determinant is Zero

To find when the matrix has no inverse, set \(\frac{x^2}{x-1} = 0\). Solving \(x^2 = 0\) gives \(x = 0\). However, since the expression has a denominator \(x-1\), check that \(x eq 1\) (which would make the denominator zero and hence undefined). Thus, the matrix has no inverse if \(x = 0\) or \(x = 1\).

Step 5: Conclusion on the Inverse

The matrix is non-invertible for any \(x\) that makes the determinant zero or undefined. Thus, when \(x = 0\) or \(x = 1\), the matrix does not have an inverse.

Key Concepts

Determinant of a Matrix

In linear algebra, the determinant is essentially a scalar value that can be computed from the elements of a square matrix. This value offers important information about the matrix, one of which is determining if a matrix is invertible. For a 2x2 matrix, the determinant is calculated using a simple formula:

• If we have a matrix \( \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \), then the determinant \( \det \) is \( ad - bc \).
• This value decisions if the matrix has an inverse; if and only if the determinant is non-zero, the matrix will have an inverse.

Calculating the determinant of the given matrix, \( \begin{bmatrix} x & 1 \ -x & \frac{1}{x-1} \end{bmatrix} \), requires multiplying \( x \) by \( \frac{1}{x-1} \) and \(-1\) by \(-x\), then subtracting these results. After simplifying, the determinant becomes \( \frac{x^2}{x-1} \). This expression helps us understand the invertibility conditions of the matrix effectively.

Non-Invertible Matrix

A non-invertible, or singular, matrix is one that does not possess an inverse. This occurs when the determinant of the matrix is zero, full stop. The disappearance of the matrix's inverse means certain solutions to equations might not be possible, making non-invertible matrices crucial to identify. In the exercise given, determining when our matrix does not have an inverse stems directly from understanding when our determinant is either zero or undefined. Given the simplified determinant expression \( \frac{x^2}{x-1} \), we see it becomes zero when the expression numerator (\(x^2\)) is zero, thus \(x = 0\). However, we also must avoid situations where the determinant calculation doesn't work due to division by zero. That's when \(x = 1\), as it makes the denominator zero and hence undefined. Therefore, the matrix is viewed as non-invertible when \(x = 0\) or \(x = 1\).

Condition for Matrix Inverse

To achieve an inverse for a matrix, certain conditions must be met. For a 2x2 matrix, this primarily means ensuring the determinant is not zero. The concept boils down to:

• If the determinant \( ad - bc = 0 \), there is no inverse available for the matrix.
• If the determinant is anything else than zero, an inversion process can occur, specifically calculated through a set formula.

In this task, determining when the matrix \( \begin{bmatrix} x & 1 \ -x & \frac{1}{x-1} \end{bmatrix} \) can have no inverse required checking the determinant condition for zero or undefined values to happen, as analyzed in our solution. Recognizing before attempting an inversion when these conditions apply can save time and ensure proper mathematical understanding. This process helps clarify why sometimes matrices might look intimidating but, with determinants, those resembling a simple check, all can be handled smoothly.