Problem 11 - Precalculus: Mathematical for Calculus

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 has no inverse, as its determinant is 0.

Step 1: Determine the Matrix Notation

Identify the matrix to be \[A = \left[\begin{array}{rr}6 & -3 \ -8 & 4\end{array}\right]\]

Step 2: Calculate the Determinant of the Matrix

The formula to find the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \(ad - bc\). For matrix \(A\), calculate\[6 \times 4 - (-3) \times (-8) = 24 - 24 = 0.\]

Step 3: Analyze the Determinant Value

Since the determinant is 0, the matrix \(A\) is singular, which means it does not have an inverse. A matrix must have a non-zero determinant to possess an inverse.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can help us understand certain properties of the matrix. For a 2x2 matrix \ \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula \ \( ad - bc \). This number can tell us whether the matrix can be inverted or not.

In the original exercise, we dealt with a 2x2 matrix \ \( \begin{bmatrix} 6 & -3 \ -8 & 4 \end{bmatrix} \). When calculating its determinant, we follow the formula:

• Multiply the diagonals: \(6 \times 4\) and \(-3 \times -8\)
• Subtract the second product from the first: \(24 - 24\)
• The determinant is found to be \(0\)

A determinant of zero has significant implications, as we will explore later in this article.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices, consisting of two rows and two columns. It looks like \ \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), where \(a\), \(b\), \(c\), and \(d\) are elements of the matrix.

Such matrices are frequently used in basic computations in linear algebra due to their simplicity. Calculations like finding the determinant or performing matrix multiplication require fewer steps than larger matrices. This makes them ideal for learning and understanding foundational matrix operations.

In our exercise, we used a 2x2 matrix to try and find its inverse, highlighting the importance of understanding how each element contributes to the overall properties of the matrix.

Singular Matrix

A singular matrix is a matrix that does not have an inverse. One major indicator that a matrix is singular is when its determinant is zero. This happens because matrices need a non-zero determinant to be invertible.

In the given exercise, the matrix \ \( \begin{bmatrix} 6 & -3 \ -8 & 4 \end{bmatrix} \) was found to have a determinant of zero. This tells us that the matrix is singular and cannot be inverted.

Why is this important? Because in many mathematical and applied fields, finding an inverse is crucial for solving systems of equations or inverting functions. An inverse matrix can "undo" the transformation applied by the original matrix, so without it, many of these operations cannot be performed as intended. Understanding singular matrices can, therefore, be essential for ensuring accuracy in various computations.