Problem 6 - Precalculus: Functions and Graphs

Question

Exer. \(3-12:\) Find the inverse of the matrix if it exists. $$\left[\begin{array}{ll} 3 & -1 \\ 6 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix does not have an inverse.

Step 1: Determine if the Matrix is Invertible

To find the inverse of a matrix, we must first calculate its determinant. If the determinant is zero, the matrix is not invertible. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \). For our matrix, \( a = 3 \), \( b = -1 \), \( c = 6 \), and \( d = -2 \), so the determinant is \( 3(-2) - (-1)(6) = -6 + 6 = 0 \).

Step 2: Conclude the Matrix is Singular

Since the determinant of the matrix is zero, it means that the matrix is singular and does not have an inverse.

Key Concepts

Determinants

A determinant is a special number associated with a square matrix, pivotal in determining some of its key properties. It provides crucial information about the matrix, such as whether it is invertible or singular. For a 2x2 matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix},\] the determinant is calculated using the formula: \[det(A) = ad - bc\]. Understanding this formula is essential when working with small matrices as it acts as a "test" to see if we can find their inverses or solve systems of linear equations that use them.

• If the determinant is zero, the matrix is singular and doesn't have an inverse.
• If the determinant is non-zero, the matrix is invertible.

Keep in mind that determinants are only defined for square matrices, and their calculation becomes more complex as the size of the matrix increases. Yet, for 2x2 matrices, it remains an easy-to-calculate and powerful tool for quick mathematical analysis.

Singular Matrices

Singular matrices play a crucial role in linear algebra. They are those special matrices that lack an inverse. A matrix is termed 'singular' if its determinant equals zero. This characteristic renders them non-invertible; hence, there's no matrix you can multiply it by to yield the identity matrix. Understanding singular matrices is important in various applications:

• In solving systems of equations, a singular matrix signifies that the system either has no solution or infinitely many solutions.
• In geometry, singular matrices often correspond to transformations that "flatten" space, such as collapsing a three-dimensional shape into a two-dimensional plane.

Recognizing singular matrices can save time in computations by avoiding the pursuit of an inverse that doesn't exist. It also indicates that something interesting or special is happening in the context of a mathematical problem.

2x2 Matrices

2x2 matrices are among the simplest forms of matrices and serve as a fundamental building block in learning matrix algebra. Their straightforward nature makes them an ideal starting point for understanding complex operations that can be extended to larger matrices. A 2x2 matrix looks like this: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]. With these matrices, many operations, including addition, multiplication, and even finding determinants, are easier to perform than with larger matrices. For instance:

• Addition: Add corresponding elements from each matrix.
• Multiplication: Use the dot product of rows and columns.
• Determinant: Compute it using the simple formula \( ad - bc \).

2x2 matrices are instrumental in initial learning stages because they allow students to get hands-on experience with fundamental concepts. When examining the properties of these matrices, like invertibility, calculations are relatively straightforward, reinforcing core matrix ideas before tackling higher dimensions.