Problem 11 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. \(\left[\begin{array}{rr}{6} & {-3} \\ {-8} & {4}\end{array}\right]\)

Step-by-Step Solution

Answer

The matrix does not have an inverse as its determinant is zero.

Step 1: Check if the inverse exists

To find the inverse of a matrix, it must be non-singular. This means its determinant should not be zero. The determinant of the matrix \( A = \begin{bmatrix} 6 & -3 \ -8 & 4 \end{bmatrix} \) is calculated as \( \text{det}(A) = (6 \times 4) - (-3 \times -8) \).

Step 2: Calculate the determinant

Calculate \( \text{det}(A) = 24 - 24 = 0 \). Since the determinant is zero, the matrix\( A \) is singular and does not have an inverse.

Step 3: Conclusion: Inverse does not exist

Given that the determinant is zero, the matrix \( A \) is singular. Therefore, it does not have an inverse.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. Essentially, it's a scalar value that provides insights into the matrix's properties, such as invertibility. To find the determinant of a 2x2 matrix, you can use the formula: \[ \text{det}(A) = ad - bc \]where \(A\) is the matrix:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\].
In our original exercise, the matrix given is \[\begin{bmatrix} 6 & -3 \ -8 & 4 \end{bmatrix}\], and the determinant is calculated as:

• Multiply the elements \(6\) and \(4\), which equals \(24\).
• Multiply the elements \(-3\) and \(-8\), which also equals \(24\).
• Subtract these products: \(24 - 24 = 0\).

This result lets us know whether the matrix has an inverse.

Singular Matrix

A singular matrix is one that does not have an inverse. For a matrix to be invertible, its determinant must be non-zero. A singular matrix, therefore, has a determinant of zero. In our exercise, we found that \( \text{det}(A) = 0 \), indicating that the matrix is singular.
This happens because the rows or columns of the matrix are linearly dependent. In simple terms, it means one row or column is just a multiple of the other. This property makes the system represented by the matrix unable to solve uniquely for variables, hence the lack of an inverse.
Understanding this aspect helps us know if we can find an inverse or if the matrix needs modification for further operations.

Matrix Algebra

Matrix algebra involves the study and manipulation of matrices to solve equations, transform data, and perform a variety of computations in fields like physics, engineering, and computer science. It encompasses a wide range of operations, including addition, subtraction, multiplication, and finding inverses.
The primary aim of matrix algebra is to express and solve linear equations efficiently. Inverse operations are particularly important as they help find solutions to matrix equations. However, not all matrices have inverses. The determinant plays a crucial role in matrix algebra since it informs us about the invertibility of a matrix.
Learning matrix algebra isn't just about manipulating numbers. It's also about understanding the theory behind these numbers, such as why a determinant of zero indicates no inverse. This understanding provides foundational knowledge for advanced mathematical concepts and practical applications.