Problem 13 - 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 is not invertible because the determinant is zero.

Step 1: Compute the determinant

\(A = \begin{bmatrix} 6 & -3 \\ -8 & 4 \end{bmatrix}\)

\(\det(A) = (6)(4) - (-3)(-8) = 24 - 24 = 0\)

Step 2: Determine invertibility

Since \(\det(A) = 0\), the matrix is \(\textbf{singular}\) (not invertible). The inverse does not exist.

Key Concepts

determinant of a matrix

The determinant is a crucial value that can be computed from a square matrix. For a 2x2 matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]the determinant is calculated using the formula \(ad - bc\). This value helps in understanding properties of the matrix, such as whether it is invertible or not. If the determinant equals zero, the matrix is referred to as singular, meaning it does not have an inverse. Conversely, if the determinant is non-zero, the matrix is non-singular and invertible.
Calculating determinants is easier for 2x2 matrices compared to larger ones, making it a fundamental concept in linear algebra.

invertible matrix

An invertible matrix, also known as a non-singular matrix, is one that has an inverse. The inverse of a matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\)exists if and only if its determinant \( ad - bc \) is not zero. When a matrix is invertible, multiplying it by its inverse results in the identity matrix \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).
For 2x2 matrices, the formula to find the inverse is: \[\frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]As seen in our example, if the determinant is zero, the matrix is not invertible, and this formula cannot be used. Having an invertible matrix is crucial for solving systems of equations, finding transformations and other applications in math.

2x2 matrix

A 2x2 matrix is one of the simplest forms of matrices and forms the starting point for learning linear algebra. It has two rows and two columns. Typically written as:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]each element can be a real number, and matrices can represent various models such as systems of linear equations, transformations, or even network flows. These matrices are easy to manipulate mathematically, with operations such as addition, subtraction, and multiplication straightforward to perform compared to larger matrices.
Thus, understanding 2x2 matrices sets a foundation for more advanced topics in linear algebra.

linear algebra concepts

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear transformations, and matrices. It plays a vital role in various fields such as engineering, physics, computer science, and economics.
Key concepts include understanding how matrices can represent systems of equations and how they transform space. Inverting matrices is another fundamental concept allowing for solving these systems, provided the matrices in question are invertible.
Determinants and the ability to identify inverses further enable mathematics students and professionals to explore deeper relationships within vectors and systems. By mastering these concepts, one gains a powerful toolkit for analyzing and solving complex mathematical problems in both theoretical and practical applications.