Problem 45 - Algebra and Trigonometry

Question

Find the inverse of the matrix. For what value(s) of x, if any, does the matrix have no inverse? $$ \left[\begin{array}{rrr}{1} & {e^{x}} & {0} \\ {e^{x}} & {-e^{2 x}} & {0} \\\ {0} & {0} & {2}\end{array}\right] $$

Step-by-Step Solution

Answer

The matrix is invertible for all real \(x\), as the determinant is never zero.

Step 1: Determine the Determinant

To find the inverse of a matrix, first compute its determinant. For this 3x3 matrix, \[\text{det}(A) = 1 \cdot \left( -e^{2x} \cdot 2 - 0 \right) - e^x \cdot \left(e^x \cdot 2 - 0 \right) + 0 \cdot \left(e^x \cdot 0 - (-e^{2x}) \cdot 0\right) = -2e^{2x} - 2(e^x)^2.\] This simplifies to \[\text{det}(A) = -2e^{2x} - 2e^{2x} = -4e^{2x}.\] Since the determinant is zero when \[-4e^{2x} = 0,\] this occurs when \[e^{2x} = 0.\] However, \(e^{2x}\) is never zero for any real \(x\). Therefore, the matrix has a determinant of zero for no real value of \(x\).

Step 2: Find the Inverse Matrix Condition

Since the determinant is never zero for real \(x\), the matrix is always invertible. To find the inverse, use the formula for the inverse of a 3x3 matrix: \[A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A),\] where \(\text{adj}(A)\) is the adjugate of \(A\). However, calculating \(A^{-1}\) requires a nonzero determinant, so we confirm invertibility.

Step 3: Conclusion on Invertibility

As we previously found, \[\text{det}(A) = -4e^{2x},\] which is never zero for real \(x\), indicating that the matrix has an inverse for all real \(x\) values, since the exponential function never yields zero.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number calculated from its elements. It plays a crucial role in understanding whether a matrix is invertible. For any 2x2 or 3x3 matrix, the determinant can indicate important properties about the matrix.

When dealing with a 3x3 matrix, to find the determinant, we use a specific formula that involves multiplying and adding or subtracting the elements and their cofactors. For the given matrix:

• The element combinations are used in a cross-multiplying fashion, similar to a mix of multiplication and addition involving all rows.
• This complexity is why understanding how to arrange and handle cofactors is important for solving the determinant.

The determinant is essential because it helps determine if we can find an inverse of the matrix. If the determinant equals zero, the matrix won't be invertible. In practice, knowing how to calculate it correctly is the first step in checking invertibility.

Adjugate of a Matrix

The adjugate of a matrix is intimately connected to finding a matrix inverse. The adjugate is essentially the transpose of the cofactor matrix derived from the original matrix.

Here's how you find it:

• First, calculate the cofactors for each element of the matrix, which involves the determinant of the submatrix formed by removing the row and column of the specific element.
• The cofactor matrix is then formed by applying the right signs according to a checkerboard pattern of positives and negatives.
• After forming the cofactor matrix, you take its transpose – this matrix becomes the adjugate.

This adjugate matrix, combined with the determinant, allows the computation of the inverse using the formula: \[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]Understanding and constructing the adjugate is a valuable skill in linear algebra, as it provides a practical way to tackle matrix inversion when the determinant is not zero.

Invertibility of Matrices

Invertibility is a core concept in linear algebra. A matrix is invertible if there exists another matrix that when multiplied results in the identity matrix, much like how with regular numbers, every number has a reciprocal except zero.

For matrices, invertibility hinges on the determinant:

• If a matrix has a determinant of zero, then the matrix does not have an inverse. It is called "singular".
• If a matrix's determinant is anything but zero, it is called "nonsingular" or "invertible". It assures that an inverse matrix exists.

In our specific problem, the determinant \( -4e^{2x} \) is never zero for any real \( x \) because the exponential function \( e^{2x} \) is always positive, which confirms the matrix is always invertible.

This concept of invertibility has widespread applications, such as solving systems of linear equations, where the presence of an inverse matrix can simplify solutions significantly.