Problem 43 - Algebra and Trigonometry

Question

Find \(A^{-1}\) and check. $$ A=\left[\begin{array}{cc} {e^{x}} & {e^{3 x}} \\ {-e^{3 x}} & {e^{5 x}} \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is zero. Hence, the inverse of the matrix does not exist.

Step 1: Compute the Determinant

The determinant (denoted as \(Det\)) of a 2x2 matrix \(A = \left[\begin{array}{cc}a & b \ c & d\end{array}\right]\) is computed as \(Det = (a * d) - (b * c)\). For our matrix, \(Det = e^{x} * e^{5x} - e^{3x} * -e^{3x} = e^{6x} - e^{6x} = 0\).

Step 2: Check the Determinant

Since \(Det = 0\), the inverse of the matrix does not exist as per the formula in step 1.

Key Concepts

Determinant of a Matrix

Understanding the determinant of a matrix is crucial for topics such as matrix inverses, system of equations, and linear transformations. The determinant provides information about the matrix that is essential in various mathematical processes.
For a 2x2 matrix \(A\) represented as \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is computed using the formula \(Det = ad - bc\).
It's easy to remember this formula as you multiply diagonally: the product from top left to bottom right minus the product from top right to bottom left.

• A non-zero determinant indicates the matrix is invertible.
• A zero determinant tells us that the matrix is singular, meaning no inverse exists.

Given the matrix \(\begin{bmatrix} e^x & e^{3x} \ -e^{3x} & e^{5x} \end{bmatrix}\), we calculate the determinant to be \(e^{6x} - e^{6x} = 0\). Here, since the determinant is zero, it confirms this matrix does not have an inverse.

2x2 Matrix

A 2x2 matrix is a simple yet powerful construct in linear algebra. It consists of two rows and two columns, making it one of the most fundamental matrices.
2x2 matrices are often used to represent basic linear transformations, solve linear equation systems, and more. They are compact but pose a range of possibilities in calculations and proofs.
To grasp 2x2 matrices better:

• Understand that basic operations like addition, subtraction, and multiplication are still applicable.
• Pay attention to the orientation and dimensions: always verify that dimensions agree for operations.

For the matrix \(\begin{bmatrix} e^x & e^{3x} \ -e^{3x} & e^{5x} \end{bmatrix}\), it encompasses those essential attributes: two variables raised to varied powers, clearly portraying behavior in a 2x2 framework. Even in this configuration, it is important to always check the determinant for invincibility due to the fundamental relation between a determinant and the matrix’s invertibility.

Matrix Inverses

Matrix inverses are fundamentally important in solving systems of linear equations and transforming spaces. Simply put, the inverse of a matrix \(A\) is another matrix \(A^{-1}\) such that \(A \cdot A^{-1} = I\), where \(I\) is the identity matrix.
However, not all matrices have inverses. A necessary condition for the inverse to exist is that the determinant of the matrix must be non-zero.
Here's why the inverse is useful:

• Solving \(Ax = b\) can be simplified to \(x = A^{-1}b\), providing solutions more directly.
• In multidimensional transformations, inverses ensure you can return to the original configuration.

For the matrix \(\begin{bmatrix} e^x & e^{3x} \ -e^{3x} & e^{5x} \end{bmatrix}\), since the determinant is zero, the matrix does not have an inverse. This highlights the determinant's role and why understanding it is key in determining invertibility.