Problem 44 - Precalculus Mathematics for Calculus

Question

Find the inverse of the matrix. For what value(s) of \(x\) if any, does the matrix have no inverse? $$\left[\begin{array}{cc} e^{x} & -e^{2 x} \\ e^{2 x} & e^{3 x} \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix always has an inverse; no value of \(x\) makes it singular.

Step 1: Calculate the Determinant

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is given by \( ad - bc \). For the given matrix, substitute \( a = e^x \), \( b = -e^{2x} \), \( c = e^{2x} \), \( d = e^{3x} \). The determinant is calculated as follows: \[ \det(A) = e^x \cdot e^{3x} - (-e^{2x}) \cdot e^{2x} = e^{4x} + e^{4x} = 2e^{4x} \].

Step 2: Determine Condition for Invertibility

A matrix is invertible if and only if its determinant is non-zero. Thus, we need to find for which values of \( x \) the determinant \( 2e^{4x} = 0 \). Since the exponential function \( e^{4x} \) is never zero for any real number \( x \), \( 2e^{4x} \) is also never zero. Therefore, the matrix always has an inverse and there are no values of \( x \) that make the determinant zero.

Step 3: Calculate the Inverse Matrix

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), if the determinant \( \det(A) = ad - bc eq 0 \), the inverse is given by:\[ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \].Given the determinant \( 2e^{4x} \), the inverse of the matrix is:\[ A^{-1} = \frac{1}{2e^{4x}} \begin{bmatrix} e^{3x} & e^{2x} \ -e^{2x} & e^x \end{bmatrix} \]. This simplifies to:\[ A^{-1} = \begin{bmatrix} \frac{e^{-x}}{2} & \frac{e^{-2x}}{2} \ -\frac{e^{-2x}}{2} & \frac{e^{-3x}}{2} \end{bmatrix} \].

Key Concepts

Determinant of a Matrix

In the world of matrices, the determinant is a crucial number. It helps us understand the properties of a square matrix. The determinant is like a fingerprint. It tells us whether a matrix is invertible. For a 2x2 matrix, calculating this fingerprint is straightforward.

The formula is simple: for a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is \( ad - bc \). This value helps in identifying a matrix's characteristics, particularly its invertibility. If the determinant is not zero, the matrix has an inverse.

Let's consider the matrix from our exercise. Given as \( \begin{bmatrix} e^{x} & -e^{2x} \ e^{2x} & e^{3x} \end{bmatrix} \), the determinant is calculated as follows:

• Multiply \( e^x \) and \( e^{3x} \) resulting in \( e^{4x} \)
• Multiply \( -e^{2x} \) and \( e^{2x} \) resulting in \( -e^{4x} \)
• Add these results: \( e^{4x} + e^{4x} = 2e^{4x} \)

Thus, the determinant \( 2e^{4x} \) plays a critical role here.

Invertibility and Its Conditions

A matrix's invertibility depends heavily on its determinant. If the determinant equals zero, the matrix is classified as non-invertible or singular. In simple terms, a singular matrix poses a dead-end; there is no path to find its inverse.

For our specific matrix, we found its determinant to be \( 2e^{4x} \). Let's break down how the exponential function impacts this. The exponential function \( e^{4x} \) is never zero for any real number. Its unique property ensures that it always produces a positive value. Remember, an exponential function continuously grows or decays but never becomes zero.

• A non-zero determinant means the matrix is invertible.
• The function \( 2e^{4x} \) can't be zero, affirming the matrix has an inverse for all real \( x \).

This property underscores why the matrix's invertibility doesn't depend on \( x \).

Exponential Function Influence on Matrix Properties

The exponential function \( e^x \) is a vital element in many mathematical scenarios. It's defined as a function that grows at an exponential rate. This function has a pivotal role in matrix algebra related to our problem.

Understanding the behavior of \( e^x \) can help us with equations that rely on this foundation. Regardless of \( x \), \( e^x \) produces a positive number, ensuring it never hits zero. Because it never reaches zero, any expression that includes \( e^x \) keeps afloat in non-zero territory, which is critical in finding inverses.

• An exponential function in the form \( e^{4x} \) impacts the determinant significantly.
• If \( e^{4x} \) were zeros, the determinant would crumble, leading to non-invertibility.

Thus, the perpetual positivity of \( e^{4x} \) ensures the matrix remains invertible, regardless of \( x \). This fascinating interplay showcases the robustness and reliability of exponential functions in matrix mathematics.