Problem 37 - Advanced Engineering Mathematics

Question

$$ \text { Show that if } \mathbf{A} \text { is a nonsingular matrix, then } \operatorname{det} \mathbf{A}^{-1}=1 / \operatorname{det} \mathbf{A} \text {. } $$

Step-by-Step Solution

Answer

\( \operatorname{det}(\mathbf{A}^{-1}) = \frac{1}{\operatorname{det}(\mathbf{A})} \) since \( \mathbf{A} \) is non-singular.

Step 1: Understanding Non-singular Matrices

A matrix \( \mathbf{A} \) is called non-singular if it has an inverse. This implies that when \( \mathbf{A} \) is multiplied by its inverse \( \mathbf{A}^{-1} \), the result is the identity matrix, \( \mathbf{I} \). Thus, \( \mathbf{A} \times \mathbf{A}^{-1} = \mathbf{I} \).

Step 2: Using Determinant Properties

One important property of determinants is that the determinant of a product of matrices is the product of their determinants. Therefore, for \( \mathbf{A} \times \mathbf{A}^{-1} = \mathbf{I} \), the determinant can be written as \( \operatorname{det}(\mathbf{A} \times \mathbf{A}^{-1}) = \operatorname{det}(\mathbf{I}) \).

Step 3: Determinant of the Identity Matrix

The determinant of the identity matrix \( \mathbf{I} \) is always 1, because it has 1s along its diagonal and 0s elsewhere, giving \( \operatorname{det}(\mathbf{I}) = 1 \).

Step 4: Applying Determinant Property

From Step 2, we have \( \operatorname{det}(\mathbf{A} \times \mathbf{A}^{-1}) = \operatorname{det}(\mathbf{A}) \cdot \operatorname{det}(\mathbf{A}^{-1}) \). Setting this equal to \( \operatorname{det}(\mathbf{I}) = 1 \), we get \( \operatorname{det}(\mathbf{A}) \cdot \operatorname{det}(\mathbf{A}^{-1}) = 1 \).

Step 5: Solving for the Determinant of the Inverse

To solve for \( \operatorname{det}(\mathbf{A}^{-1}) \), divide both sides of the equation by \( \operatorname{det}(\mathbf{A}) \) (assuming \( \operatorname{det}(\mathbf{A}) eq 0 \) as \( \mathbf{A} \) is non-singular), giving \( \operatorname{det}(\mathbf{A}^{-1}) = \frac{1}{\operatorname{det}(\mathbf{A})} \).

Key Concepts

Nonsingular Matrix

A nonsingular matrix is one that can be inverted, meaning there exists another matrix called the inverse that can multiply the original to yield the identity matrix. The identity matrix, denoted as \( \mathbf{I} \), has 1s on the diagonal and 0s elsewhere. For a matrix \( \mathbf{A} \) to be nonsingular, it must satisfy the equation \( \mathbf{A} \times \mathbf{A}^{-1} = \mathbf{I} \). This property ensures that the determinant of the matrix, \( \operatorname{det}(\mathbf{A}) \), is not zero.

• A nonsingular matrix has a unique inverse.
• The determinant of a nonsingular matrix is non-zero.
• It plays a crucial role in solving linear equations, where systems with nonsingular matrices have unique solutions.

Understanding these properties can help in verifying whether a matrix is nonsingular, and thus, invertible.

Inverse of a Matrix

The inverse of a matrix \( \mathbf{A} \), denoted as \( \mathbf{A}^{-1} \), is a matrix that can "reverse" the effects of \( \mathbf{A} \) when multiplied by it. This inverse transforms \( \mathbf{A} \) back into the identity matrix.
For example, if you have a matrix \( \mathbf{A} \) and its inverse \( \mathbf{A}^{-1} \), their multiplication results in the identity matrix: \( \mathbf{A} \times \mathbf{A}^{-1} = \mathbf{I} \). This same result also applies vice versa: \( \mathbf{A}^{-1} \times \mathbf{A} = \mathbf{I} \).
If you're working with the determinants, an interesting property comes into play—\( \operatorname{det}(\mathbf{A}^{-1}) \) is the reciprocal of \( \operatorname{det}(\mathbf{A}) \). This is fundamental in cases where finding the inverse is necessary for calculations, particularly in solving matrix equations like \( \mathbf{A}\mathbf{x} = \mathbf{b} \).
The process of finding the inverse involves several methods like Gaussian elimination, adjoint matrices, or row reduction, but it is only possible if the matrix is nonsingular.

Properties of Determinants

Determinants provide crucial information regarding a matrix's properties, notably when it comes to understanding the invertibility of a matrix. Certain properties of determinants can greatly simplify matrix calculations:

• One of these properties, as previously mentioned, is that the determinant of the product of two matrices equals the product of their determinants. This means for matrices \( \mathbf{A} \) and \( \mathbf{B} \), we have \( \operatorname{det}(\mathbf{A} \times \mathbf{B}) = \operatorname{det}(\mathbf{A}) \times \operatorname{det}(\mathbf{B}) \).
• The determinant of the identity matrix, \( \mathbf{I} \), is always 1.
• Important for inverses, the determinant helps conclude that \( \operatorname{det}(\mathbf{A}^{-1}) = \frac{1}{\operatorname{det}(\mathbf{A})} \).
• A zero determinant points to a singular matrix with no inverse.

Utilizing these properties is key in simplifying problem-solving and verifying matrix characteristics. They support fundamental tasks like checking if a system of equations is solvable.