https://file.aiquickdraw.com/mathbot/file/1722101972385vhfjgdts.PNG

To solve the given matrix equation \(AX = B\), we need to find the values of \(x\) and \(y\) by inverting matrix \(A\) and multiplying it by matrix \(B\).

Given the matrices:
\[ A = \begin{bmatrix} 2 & -4 \\ 3 & -5 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} 4 \\ 7 \end{bmatrix} \]

First, we find the inverse of matrix \(A\), \(A^{-1}\). The formula for the inverse of a 2x2 matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is:
\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]

For matrix \(A\):
\[ a = 2, \quad b = -4, \quad c = 3, \quad d = -5 \]

Calculate the determinant \(ad - bc\):
\[ \text{det}(A) = (2 \cdot -5) - (-4 \cdot 3) = -10 + 12 = 2 \]

Now, find \(A^{-1}\):
\[ A^{-1} = \frac{1}{2} \begin{bmatrix} -5 & 4 \\ -3 & 2 \end{bmatrix} = \begin{bmatrix} -\frac{5}{2} & 2 \\ -\frac{3}{2} & 1 \end{bmatrix} \]

Next, multiply \(A^{-1}\) by \(B\):
\[ X = A^{-1}B = \begin{bmatrix} -\frac{5}{2} & 2 \\ -\frac{3}{2} & 1 \end{bmatrix} \begin{bmatrix} 4 \\ 7 \end{bmatrix} \]

Perform the matrix multiplication:
\[ \begin{bmatrix} -\frac{5}{2} & 2 \\ -\frac{3}{2} & 1 \end{bmatrix} \begin{bmatrix} 4 \\ 7 \end{bmatrix} = \begin{bmatrix} -\frac{5}{2} \cdot 4 + 2 \cdot 7 \\ -\frac{3}{2} \cdot 4 + 1 \cdot 7 \end{bmatrix} = \begin{bmatrix} -10 + 14 \\ -6 + 7 \end{bmatrix} = \begin{bmatrix} 4 \\ 1 \end{bmatrix} \]

Therefore, the solution is:
\[ x = 4, \quad y = 1 \]

Do you want details or have any questions? Here are five questions you might ask next:
1. How do you determine the determinant of a 3x3 matrix?
2. Can you explain how to find the inverse of a 3x3 matrix?
3. How can matrix operations be used to solve systems of equations with more variables?
4. What are the properties of matrix inverses?
5. How is the determinant of a matrix related to its invertibility?

Tip: When finding the inverse of a matrix, always check the determinant first. If the determinant is zero, the matrix does not have an inverse.

Learn how to solve the matrix equation AX = B using matrix inversion and multiplication. This problem involves matrix algebra, determinant calculation, and matrix operations. Suitable for undergraduate level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation