Problem 79 - Precalculus with Limits

Question

WRITING Explain how to determine whether the inverse of a \(2 \times 2\) matrix exists. If so, explain how to find the inverse.

Step-by-Step Solution

Answer

The existence of the inverse of a \(2 \times 2\) matrix is determined by calculating the determinant, which should not be zero. If the determinant is non-zero, the inverse can be found using the formula \(A^{-1} = \frac{1}{ad - bc}\begin{bmatrix}d & -b\\ -c & a\end{bmatrix}\).

Step 1: Determining the Existence of the Inverse

The first step is to check whether the inverse of a given \(2 \times 2\) matrix exists. This can be determined by calculating the determinant of the matrix. For a matrix \(A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}\), the determinant \(det(A)\) is equal to \(ad - bc\). If the determinant is not equal to zero, then the inverse of the matrix exists.

Step 2: Finding the Inverse

If the determinant of the matrix \(A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}\) is not zero (\(det(A) \neq 0\)), then the inverse of the matrix \(A^{-1}\) can be found using the formula \(A^{-1} = \frac{1}{ad - bc}\begin{bmatrix}d & -b\\ -c & a\end{bmatrix}\).

Key Concepts

Determinant of a 2x2 Matrix

In a matrix, the determinant is a special number that gives us important insight into the matrix's properties. For a 2x2 matrix, calculating the determinant is straightforward and can be done using a simple formula. Consider a matrix \(A\) with elements arranged as follows: \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\] The determinant of this matrix, denoted as \(det(A)\), is calculated by taking the products of the diagonal elements and subtracting the products of the off-diagonal elements, like this: \[det(A) = ad - bc\]This calculation is essential because it helps us in deciding whether a matrix can have an inverse or not. If the determinant is zero, the matrix cannot be inverted.

Existence of Matrix Inverse

Now that we understand how to determine the determinant of a 2x2 matrix, let's discuss what it tells us about the inverse. To check if a matrix has an inverse, we simply need to ensure that its determinant is not equal to zero. If the determinant of the matrix \(A\) is zero, then the matrix is said to be singular and does not have an inverse. This is because a determinant of zero indicates that the matrix does not have full rank, or in other words, the rows or columns of the matrix are linearly dependent. On the other hand, if the determinant is non-zero, it confirms that the matrix is invertible. Once we know that the matrix can have an inverse, we can proceed to find it by using the determinant and other elements of the matrix.

Finding the Inverse of a 2x2 Matrix

Once we've confirmed that the determinant is non-zero, we can find the inverse of the matrix. The process involves rearranging and scaling the original elements of the matrix \(A\). For the matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), its inverse \(A^{-1}\) is calculated using the following formula:\[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]Here’s what we do:

• Swap the positions of \(a\) and \(d\).
• Change the signs of \(b\) and \(c\).
• Divide the entire matrix by the determinant \(ad - bc\), which is now confirmed to be non-zero.

This formula utilizes the determinant in two ways – it confirms the existence of the inverse, and it scales the transformed matrix to give us the precise inverse. Knowing how to calculate the inverse is crucial in solving systems of equations and transforming geometric entities.