Problem 92 - Precalculus

Question

Explain the mistake that is made. Find the inverse of \(A\) given that \(A=\left[\begin{array}{rr}2 & 5 \\ 3 & 10\end{array}\right]\) Solution: $$A^{-1}=\frac{1}{A} \quad A^{-1}=\frac{1}{\left[\begin{array}{rr}2 & 5 \\\3 & 10\end{array}\right]}$$ Simplify. $$A^{-1}=\left[\begin{array}{ll} \frac{1}{2} & \frac{1}{5} \\\\\frac{1}{3} & \frac{1}{10}\end{array}\right]$$ This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was dividing each matrix element directly; use the inverse formula instead.

Step 1: Understand the Problem

We need to find the inverse of matrix \( A \) using the correct formula. The provided solution attempts to directly divide each entry of matrix \( A \) by 1, which is incorrect.

Step 2: Recall the Formula for Inverse of a 2x2 Matrix

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is given by \( A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \), assuming \( ad-bc eq 0 \).

Step 3: Calculate the Determinant

Calculate the determinant of matrix \( A \), which is \( ad-bc = (2)(10) - (5)(3) = 20 - 15 = 5 \). Since the determinant is not zero, the inverse exists.

Step 4: Apply the Formula for Inverse

Substitute the values into the inverse formula: \( A^{-1} = \frac{1}{5} \begin{bmatrix} 10 & -5 \ -3 & 2 \end{bmatrix} \).

Step 5: Simplify the Matrix

Multiply each element of the matrix by \( \frac{1}{5} \) to find the inverse: \[ A^{-1} = \begin{bmatrix} 2 & -1 \ -\frac{3}{5} & \frac{2}{5} \end{bmatrix} \].

Step 6: Explain the Incorrect Approach

The mistake was trying to compute the inverse by dividing each element of the matrix by 1, which is incorrect. Instead, the formula for the inverse involves a specific calculation based on the determinant and rearrangement of matrix elements.

Key Concepts

2x2 matrix

When discussing matrices, one common topic is the 2x2 matrix. This is a simple mathematical construct that consists of two rows and two columns. An example of a 2x2 matrix can be written as: \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\). Here, \(a, b, c,\) and \(d\) are the entries of the matrix.
These matrices are fundamental in various calculations in linear algebra, such as finding determinants, computing inverses, and solving linear equations.

• Each entry can be a number, variable, or even a function.
• The main diagonal runs from the top left (\(a\)) to the bottom right (\(d\)).

Understanding how to manipulate these matrices is crucial for exploring further mathematical concepts.

determinant calculation

Calculating the determinant of a 2x2 matrix is a key step when working with inverses. The determinant provides a scalar value that helps determine the invertibility of the matrix. For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as: \[det(A) = ad - bc\].
This formula is quite simple and involves the diagonal elements.

• Multiply the elements of the main diagonal: \(a \times d\).
• Subtract the product of the off-diagonal elements: \(b \times c\).

If the determinant is zero, the matrix does not have an inverse. Otherwise, you can proceed to find the inverse using this determinant value.
Calculating determinants also plays a role in evaluating matrix solutions to systems of equations, making this a fundamental skill in algebra.

inverse formula

Finding the inverse of a 2x2 matrix relies on a specific formula, which is only valid if the determinant is non-zero. For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the inverse is given by: \[ A^{-1} = \frac{1}{det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \], where \(det(A) = ad - bc\).

• Switch the positions of \(a\) and \(d\).
• Negate the values of \(b\) and \(c\).
• Divide each element by the determinant \(det(A)\).

This method transforms the original matrix by rearranging its elements and scaling them by the determinant. It's important to remember this approach is distinct from simply dividing each element by a number, which is incorrect. Properly using the inverse formula requires paying attention to both the layout and the values within the matrix.