Problem 97 - Precalculus

Question

Determine whether the statements are true or false. $$\text { If } A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], \text { then } A^{-1}=\left[\begin{array}{cc}\frac{1}{a_{11}} & \frac{1}{a_{12}} \\\\\frac{1}{a_{21}} & \frac{1}{a_{22}}\end{array}\right]$$

Step-by-Step Solution

Answer

The statement is false.

Step 1: Understand the Given Matrix Structure

The given matrix \( A \) is a \( 2 \times 2 \) matrix represented as \( A = \left[ \begin{array}{cc} a_{11} & a_{12} \ a_{21} & a_{22} \end{array} \right] \). To find the inverse, \( A^{-1} \), if it exists, we will analyze the general formula for the inverse of a \( 2 \times 2 \) matrix.

Step 2: Recall Formula for 2x2 Matrix Inversion

For a \( 2 \times 2 \) matrix \( A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \), the inverse \( A^{-1} \) exists if the determinant, \( ad - bc \), is not zero. The formula for \( A^{-1} \) is:\[ A^{-1} = \frac{1}{ad - bc} \left[ \begin{array}{cc} d & -b \ -c & a \end{array} \right] \]

Step 3: Compare Given Inverse with Correct Formula

Compare the matrix provided for \( A^{-1} \) in the problem statement, \( \left[ \begin{array}{cc} \frac{1}{a_{11}} & \frac{1}{a_{12}} \ \frac{1}{a_{21}} & \frac{1}{a_{22}} \end{array} \right] \), with the correct formula from Step 2. Notice that the given inverse doesn't match the correct inverse structure because it simply contains the reciprocal of individual elements.

Step 4: Determine the Veracity of the Statement

The structure given for \( A^{-1} \) in the problem statement is incorrect as it doesn't adhere to the established formula for matrix inversion. Specifically, it lacks adjustments along both diagonals and division by the determinant \( ad - bc \).

Step 5: Conclusion

After considering all the steps, it is clear that the given definition of \( A^{-1} \) in the problem statement is not correct, showing the statement is false.

Key Concepts

Determinant

A determinant is a special number that can be calculated from a square matrix, such as a 2x2 matrix. It is a crucial component in determining whether a matrix has an inverse. For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\)the determinant is calculated using the formula:\[\text{det}(A) = ad - bc.\]Here, "a", "b", "c", and "d" are the elements of the matrix. The determinant must be non-zero for the matrix to have an inverse.

If the determinant equals zero, the matrix is called singular, meaning it does not have an inverse. Understanding how to calculate the determinant helps in solving many mathematical problems involving matrices, such as those dealing with linear algebra.

2x2 Matrix

A 2x2 matrix is a simple grid of numbers arranged in two rows and two columns. These matrices are often used in various math problems due to their simplicity and usefulness in computations. An example of a 2x2 matrix is \(\begin{bmatrix} a & b \ c & d \end{bmatrix}.\)This layout is useful when calculating determinants, solving linear equations, and finding inverse matrices.

2x2 matrices are a fundamental part of linear algebra. They serve as building blocks for larger matrices and complex matrix operations. Understanding the components and operations involved in 2x2 matrices is crucial for advancing in more complicated mathematical concepts.

Inverse Matrix Formula

To determine the inverse of a 2x2 matrix, it's essential to follow a specific formula. If you have a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\)its inverse, if it exists, is given by:\[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}.\]The existence of the inverse depends on the determinant \(ad - bc\) being non-zero.

In this formula, you see that the matrix elements are reordered and negated to create the inverse. This specific pattern must be followed to correctly compute the inverse. The concept of using an inverse matrix is critical for solving systems of linear equations where the matrix represents the coefficients.