Problem 15 - Precalculus Mathematics for Calculus

Question

Find the inverse of the matrix if it exists. $$\left[\begin{array}{rr} 0.4 & -1.2 \\ 0.3 & 0.6 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of the matrix is \( \begin{bmatrix} 1 & 2 \\ -0.5 & 0.6667 \end{bmatrix} \).

Step 1: Calculate the determinant

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by the formula \( ad - bc \). In this case, \( a = 0.4 \), \( b = -1.2 \), \( c = 0.3 \), and \( d = 0.6 \). Calculate the determinant: \[\text{Determinant} = (0.4)(0.6) - (-1.2)(0.3) = 0.24 + 0.36 = 0.6 \] The determinant is 0.6.

Step 2: Check if the inverse exists

An inverse of a matrix exists only if its determinant is non-zero. Since the determinant we calculated is 0.6, which is not zero, the matrix does have an inverse.

Step 3: Calculate the inverse of the matrix

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is given by \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Plug in the values:\[\text{Inverse} = \frac{1}{0.6} \begin{bmatrix} 0.6 & 1.2 \ -0.3 & 0.4 \end{bmatrix} \]This simplifies to:\[\begin{bmatrix} 1 & 2 \ -0.5 & 0.6667 \end{bmatrix}\]Thus, the inverse of the matrix is \( \begin{bmatrix} 1 & 2 \ -0.5 & 0.6667 \end{bmatrix} \).

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It is a key player in matrix algebra and helps determine if a matrix has an inverse, among other things. For a 2x2 matrix, which consists of four elements arranged in two rows and two columns, calculating the determinant is quite straightforward.

To find the determinant of a matrix:

• Identify the elements at positions: top-left (a), top-right (b), bottom-left (c), and bottom-right (d).
• Employ the formula: \( ad - bc \) to compute the determinant.

Using this formula, the determinant tells us whether the matrix can be inverted. A determinant value of zero indicates no inverse exists. Therefore, knowing how to calculate a determinant is an essential skill when working with matrices.

2x2 Matrix

A 2x2 matrix consists of two rows and two columns, containing four elements in total. The way these elements are arranged is crucial in determining many properties of the matrix. This specific size of matrices is easy enough to work with by hand, making them ideal for learning basic concepts of matrix algebra.

Understanding a 2x2 matrix involves knowing:

• The layout, which is typically of the form \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)
• Each element's position, which plays into operations such as addition, multiplication, and finding inverses.
• The relationship between the elements for advanced operations like determinant computation and inverses.

By consolidating these basics, solving matrix problems becomes much easier and more intuitive.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and the operations you can perform on them. It lies at the heart of many applications ranging from computer graphics to solving systems of linear equations. One important task in matrix algebra is finding the inverse of a matrix, which is essentially the matrix equivalent of the reciprocal of a number.

A few key operations in matrix algebra involve:

• Matrix addition and subtraction, which involves element-wise addition or subtraction.
• Matrix multiplication, a more complex operation that requires understanding the arrangement of rows and columns.
• Finding the inverse of a matrix, which, for 2x2 matrices, uses the previously discussed determinant to ascertain its existence.

These operations, among others, form the basis of matrix algebra, making it a valuable tool for a wide range of mathematical and practical applications.