Problem 10 - College Algebra with Modeling and Visualization

Question

Find the value of the constant \(k\) in \(A^{-1}\). A=\left[\begin{array}{ll} -2 & 5 \\ -3 & 4 \end{array}\right], \quad A^{-1}=\left[\begin{array}{ll} \frac{4}{7} & -\frac{5}{7} \\ k & -\frac{2}{7} \end{array}\right]

Step-by-Step Solution

Answer

k is \( \frac{3}{7} \).

Step 1: Determine the formula for the inverse of a 2x2 matrix

For a general matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse \( A^{-1} \) is given by \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Therefore, the first task is to compute the determinant \( ad-bc \).

Step 2: Compute the determinant of matrix A

Given the matrix \( A = \begin{bmatrix} -2 & 5 \ -3 & 4 \end{bmatrix} \), the determinant is calculated as \( (-2)(4) - (5)(-3) = -8 + 15 = 7 \).

Step 3: Find the inverse of matrix A using the determinant

Given the determinant is 7, the inverse is calculated as \( \frac{1}{7} \begin{bmatrix} 4 & -5 \ 3 & -2 \end{bmatrix} = \begin{bmatrix} \frac{4}{7} & -\frac{5}{7} \ \frac{3}{7} & -\frac{2}{7} \end{bmatrix} \).

Step 4: Compare given and calculated inverse matrices to find k

The given inverse matrix is \( \begin{bmatrix} \frac{4}{7} & -\frac{5}{7} \ k & -\frac{2}{7} \end{bmatrix} \). From our calculated inverse, the element in the position of \( k \) is \( \frac{3}{7} \). Therefore, \( k = \frac{3}{7} \).

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix, the process is straightforward and crucial when dealing with matrix inverses and algebra.
The determinant of a matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by the formula:

• \(ad - bc\)

In our example, the matrix \(A\) is \(\begin{bmatrix} -2 & 5 \ -3 & 4 \end{bmatrix}\). Plugging the values into the formula, we compute the determinant as follows:

• \((-2)(4) - (5)(-3) = -8 + 15 = 7\)

The positive value \(7\) tells us that the matrix has a valid inverse. Calculating determinants can seem abstract, but it essentially measures how much the matrix expands or compresses space.

2x2 Matrix

Matrix operations are fundamental in mathematics, and understanding the structure of a 2x2 matrix is essential for learning matrix algebra. A 2x2 matrix is structured like this: \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\).
This means it has 2 rows and 2 columns, containing four numbers in total.
These matrices are popular in solving linear equations and finding matrix inverses. Importantly, each element has a specific role in operations like addition, multiplication, and inversion.
In our given example, the 2x2 matrix \(A\) is \(\begin{bmatrix} -2 & 5 \ -3 & 4 \end{bmatrix}\), where each element relates to the deterministic formula for finding inverses.

Matrix Algebra

Matrix algebra involves a range of mathematical operations that you can perform on matrices. Inverse matrices play a major role in solving linear equations, and the ability to find the inverse, especially in 2x2 matrices, is an integral part.
To find an inverse of a 2x2 matrix, the formula is:

• \(A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\)

This formula relies heavily on the determinant \(ad-bc\) to exist; if it is zero, the matrix does not have an inverse.
In our example, we've calculated the determinant to be 7, which allows us to find the inverse:

• The inverse \(A^{-1}\) of \(A = \begin{bmatrix} -2 & 5 \ -3 & 4 \end{bmatrix}\) is \(\begin{bmatrix} \frac{4}{7} & -\frac{5}{7} \ \frac{3}{7} & -\frac{2}{7} \end{bmatrix}\).

Matrix algebra thus not only includes a technique for finding inverse matrices but also encompasses operations like addition, subtraction, and multiplication, which are essential for solving more complex problems in mathematics.