Problem 17 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. \(\left[\begin{array}{rrr}{1} & {2} & {3} \\ {4} & {5} & {-1} \\ {1} & {-1} & {-10}\end{array}\right]\)

Step-by-Step Solution

Answer

The matrix does not have an inverse because its determinant is 0.

Step 1: Calculate the Determinant

To find the inverse of a 3x3 matrix, we first need to check if it is invertible by calculating the determinant. Let's denote the matrix as \( A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & -1 \ 1 & -1 & -10 \end{bmatrix} \). The determinant of a 3x3 matrix \( A \) is calculated as: \( \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \). For matrix \( A \), this expands to: \( \text{det}(A) = 1((5)(-10) - (-1)(-1)) - 2((4)(-10) - (-1)(1)) + 3((4)(-1) - (5)(1)) \). Calculate the determinant: \( \text{det}(A) = 1(-50 - 1) - 2(-40 + 1) + 3(-4 - 5) = 1(-51) + 2(39) - 3(9) = -51 + 78 - 27 = 0 \). Since the determinant is 0, the matrix is not invertible.

Step 2: Determine the Invertibility

Since the determinant calculated in Step 1 is 0, the matrix does not have an inverse. A matrix with a determinant of 0 is called singular and does not have an inverse by definition.

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It provides useful information about the matrix and its properties. The determinant of a 3x3 matrix is calculated using a specific formula:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]Here, \(a, b, c\) are elements from the first row, and\(d, e, f, g, h, i\) are elements from the second and third rows of the matrix.

Understanding determinants is important because:

• A non-zero determinant indicates that the matrix is invertible, meaning it has an inverse.
• A zero determinant indicates that the matrix is singular, meaning it does not have an inverse.

Determinants are used in various mathematical applications like solving systems of linear equations, and understanding transformations in linear algebra. It's a key indicator of the matrix's behavior and relationships between its rows and columns.

Invertible Matrix

An invertible matrix, also known as a non-singular or non-degenerate matrix, is a square matrix that has an inverse. The inverse is another matrix which, when multiplied by the original matrix, results in the identity matrix. This identity matrix is analogous to the number 1 in basic arithmetic, where any number multiplied by 1 remains unchanged.

For a matrix \(A\), the inverse is typically denoted as \(A^{-1}\), and the relationship is expressed as:
\[ A \cdot A^{-1} = I \]where \(I\) is the identity matrix. The existence of an inverse is critical for many procedures in linear algebra, such as solving equations and transformations:

• If a matrix has an inverse, it is possible to solve equations where the matrix represents a system of equations.
• It implies that the transformation represented by the matrix is reversible.

Remember, only matrices with a non-zero determinant are invertible, which means they offer flexibility and richness in the linear transformations they can represent.

Singular Matrix

A singular matrix is one with a determinant of zero, meaning it does not have an inverse. These matrices are intriguing because they signal certain constraints or limitations in linear algebraic operations. When working with a singular matrix, it's essential to understand:

• Solve certain systems of linear equations that don't have a unique solution.
• Singular matrices often arise naturally in practical problems where dependency among data is present.

A singular matrix can appear in a transformation that squashes space into a lower dimension, losing some information in the process. This transformation can involve rotation or scaling that makes the matrix non-reversible. In computing terms, singular matrices can cause challenges in methods that rely on matrix inversion, resulting in errors or instability. Thus, identifying and understanding singular matrices is critical for preventing computational issues in advanced algorithms.