Problem 44 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The matrix is always invertible; no value of \( x \) makes it non-invertible.

Step 1: Calculate the Determinant

To find the inverse of a matrix, we first need to determine if it is invertible by calculating its determinant. The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \). In this case, the determinant of the matrix \( \begin{bmatrix} e^x & -e^{2x} \ e^{2x} & e^{3x} \end{bmatrix} \) is \( e^x \cdot e^{3x} - (-e^{2x}) \cdot e^{2x} = e^{4x} + e^{4x} = 2e^{4x} \).

Step 2: Determine When the Determinant is Zero

For a matrix to have no inverse, its determinant must be zero. Therefore, we set \( 2e^{4x} = 0 \). Since \( e^{4x} > 0 \) for all real \( x \), this equation can never be true. Hence, the determinant is never zero for any real \( x \).

Step 3: Confirm Inversion Possible for All x

Since the determinant is never zero, the matrix is invertible for all real values of \( x \). There are no values of \( x \) that will cause the matrix to be non-invertible.

Key Concepts

Determinant of a Matrix

The determinant of a matrix plays a crucial role in understanding whether the matrix is invertible. For a 2x2 matrix, like the one given in the exercise, the determinant is calculated using the formula \( ad - bc \), where \( a, b, c, \) and \( d \) are the elements of the matrix. In our specific example,

• \( a = e^x \)
• \( b = -e^{2x} \)
• \( c = e^{2x} \)
• \( d = e^{3x} \)

To find the determinant, substitute these into the formula to get:\[ e^x \cdot e^{3x} - (-e^{2x}) \cdot e^{2x} = e^{4x} + e^{4x} = 2e^{4x}. \]This result tells us that the matrix is not singular, meaning it doesn’t have a determinant of zero, which directly influences whether it has an inverse or not.

Invertible Matrices

An invertible matrix is one that can be "reversed" in a sense. For a matrix to be invertible, it must have a non-zero determinant. This is because the determinant being non-zero implies that the linear transformations associated with the matrix do not squash any dimension, allowing a unique inverse to exist.
If the determinant were zero, the matrix would be singular, which results in no inverse existing for it. In this exercise, we confirmed the determinant is \( 2e^{4x} \). Since \( e^{4x} > 0 \) for any real number \( x \), it follows that \( 2e^{4x} eq 0 \), guaranteeing that the matrix is invertible for all real \( x \).
This particular finding is crucial because it shows that there are no exclusions of \( x \) that turn the matrix into one without an inverse.

Exponential Functions

Exponential functions are a type of mathematical function where the variable is in the exponent. This is key in this exercise because each element of the matrix is defined using exponential functions of \( x \).
For instance, we have terms like \( e^x, e^{2x}, \) and \( e^{3x} \) populating the matrix. One of the important properties of exponential functions is that they are always positive for real numbers. This ensures that expressions such as \( e^{4x} \) are never zero.
Understanding this property helped us conclude that the term contributing to the determinant, \( 2e^{4x} \), was always positive. Thus, it assured us that the determinant never equates to zero for any real \( x \), reinforcing that the matrix retains its invertibility across all real values. This insight, rooted in the nature of exponential functions, is vital for resolving problems involving exponential matrices like the one in this exercise.