Problem 13 - Precalculus Mathematics 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 does not have an inverse because its determinant is 0.

Step 1: Check Determinant for Inversibility

Calculate the determinant of the matrix \( A = \left[ \begin{array}{rr} 6 & -3 \ -8 & 4 \end{array} \right] \). The formula for the determinant of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) is \( ad - bc \). Thus, for matrix \( A \), the determinant is calculated as:\[ \text{det}(A) = (6)(4) - (-3)(-8) = 24 - 24 = 0 \] Since the determinant is 0, the matrix \( A \) does not have an inverse.

Step 2: Conclude on the Existence of the Inverse

Since the determinant of matrix \( A \) is 0, matrix \( A \) is singular, meaning it does not have an inverse. A matrix must have a non-zero determinant to be invertible.

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It plays a crucial role in determining whether a matrix is invertible. For a 2x2 matrix, the formula to find the determinant is straightforward:

• For a matrix \(\left[ \begin{array}{cc}a & b \c & d\end{array} \right]\), the determinant is calculated as \(ad - bc\).

This formula involves multiplying the diagonals of the matrix and subtracting the product of the off-diagonal terms. Determinants are useful in various mathematical applications, including solving systems of linear equations and evaluating integrals. In this context, we use it to assess whether an inverse matrix exists. If the determinant is zero, the matrix lacks an inverse, making it singular.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix used in linear algebra. It consists of two rows and two columns. Here, every element in the matrix is referenced by its position color-coded into rows and columns. For example:

• \(\left[ \begin{array}{cc}6 & -3 \-8 & 4\end{array} \right]\)

This matrix can quickly demonstrate key concepts such as matrix multiplication, determinants, and inverses. For beginners, it provides a good starting point to understand more complex matrices. When performing operations on a 2x2 matrix, calculations are more manageable, and they serve as a building block for understanding larger matrices.

Understanding how to perform basic operations, such as finding the determinant, is essential before moving on to higher-dimension matrices.

Singular Matrix

A singular matrix is a square matrix that does not have an inverse. In simpler terms, this means there's no other matrix that you could multiply by the singular matrix to get the identity matrix back. For our case:

• If the determinant of a matrix is zero, it is classified as a singular matrix.

This property indicates that a set of linear equations presented by the matrix has no unique solutions. For instance, parallel lines (or dependent systems) in a linear equation representation contribute to a determinant of zero, confirming singularity. Diagrams or practical applications in physics often use this property when considering equilibrium conditions. A singular matrix reflects dependencies or redundancies between the rows or columns of the matrix, making it critical to recognize this when performing matrix operations.