Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Inverse of a \(2 \times 2\) Matrix, use the formula on page 676 to find the inverse of the \(2 \times 2\) matrix (if it exists). $$\left[\begin{array}{rr} 7 & 12 \\ -8 & -5 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of the given \(2\times2\) matrix is \(A^{-1}= \left[\begin{array}{cc} \frac{-5}{61} & \frac{-12}{61} \ \frac{8}{61} & \frac{7}{61} \end{array}\right]\)

Step 1: Identify the elements of the matrix

Given a \(2 \times 2\) matrix \(A = \left[\begin{array}{cc} a & b \ c & d \end{array}\right]\), let's identify its elements. In this case, \(a=7\), \(b=12\), \(c=-8\), and \(d=-5\).

Step 2: Calculate the determinant

Let's calculate the determinant of matrix \(A\), which is \( det(A) = ad - bc\). This equals \( (7 \times -5) - (12 \times -8) = -35 - (-96) = 61 \).

Step 3: Check if the matrix has an inverse

A matrix has an inverse if and only if its determinant is non-zero. Here the determinant of the matrix is 61, which is non-zero, and therefore the matrix does have an inverse.

Step 4: Calculate the inverse matrix

According to the formula, the inverse of matrix \(A\) is \(A^{-1} = \frac{1}{ad - bc} \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right]\). Substituting our values into this formula, we get \(A^{-1} = \frac{1}{61} \left[\begin{array}{cc} -5 & -12 \ 8 & 7 \end{array}\right]\) which simplifies to \(A^{-1}= \left[\begin{array}{cc} \frac{-5}{61} & \frac{-12}{61} \ \frac{8}{61} & \frac{7}{61} \end{array}\right]\)

Key Concepts

Determinant of a Matrix

The determinant is a scalar value that is a function of the entries of a square matrix. It provides important mathematical properties and is used to determine whether a matrix has an inverse, among other applications.

For a 2 \times 2 matrix, which is the focus of our exercise, the formula to calculate the determinant is quite straightforward. Given a matrix
\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]
the determinant is calculated using the formula
\[ det(A) = ad - bc \].

It reflects the scaling factor that the matrix applies to an area. If you imagine a parallelogram formed by the row vectors of the matrix, the determinant gives the area of that parallelogram. A determinant of zero implies that the area would be zero, meaning the matrix would collapse the space into a lower dimension, which in turn means the matrix cannot be inverted.

Non-Zero Determinant

For a matrix to have an inverse, it's crucial that its determinant is not equal to zero. This concept of a non-zero determinant is essential because it indicates that the matrix can be used for certain operations without losing or distorting information.

A non-zero determinant tells us two main things: First, the matrix, when applied to a geometric figure, does not reduce it to a lower dimensional shape. Second, it confirms the existence of a unique matrix that can 'undo' the transformation applied by the original matrix.

This is also a fundamental condition for solving systems of linear equations; if the matrix associated with the system has a non-zero determinant, you can be sure that there is a unique solution.

Matrix Inversion Formula

A matrix inversion formula for a 2 \times 2 matrix is a direct method to find the inverse of a matrix, provided that its determinant is non-zero. The formula is
\[ A^{-1} = \frac{1}{det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]
where A is the original matrix, and A^{-1} is its inverse.

In the given exercise, the inversion formula was applied stepwise to calculate the inverse of the given matrix. This inverses the action of the original matrix and is particularly useful in solving linear equations and changing coordinate systems. Due to the simplicity of this formula for 2 \times 2 matrices, it's often one of the first techniques taught to algebra students for matrix inversion.

Understanding and applying the matrix inversion formula correctly is crucial for various fields, including computer graphics, engineering, and mathematics itself.