https://file.aiquickdraw.com/mathbot/file/1726421101555p8xwvn6u.jpg

The system of matrix equations involves finding the unknown matrices \(X\) and \(Y\) using the given equations:

1. \(-5X + 6Y = A\)
2. \(4X - 5Y = B\)

Where matrices \(A\) and \(B\) are provided as:
\[
A = \begin{pmatrix} 12 & -11 \\ 0 & -15 \\ 12 & 32 \end{pmatrix}, \quad B = \begin{pmatrix} -10 & 9 \\ 0 & 12 \\ -10 & -26 \end{pmatrix}
\]

### Approach:

1. Solve the system for \(X\) and \(Y\) using linear combinations of matrices \(A\) and \(B\).
2. The form \(X = c_1 A + c_2 B\) and \(Y = d_1 A + d_2 B\) will help.
3. After determining \(X\) and \(Y\), calculate \(X^T Y\) (where \(X^T\) is the transpose of matrix \(X\)).
4. Finally, find the determinant of \(X^T Y\).

Let me compute the solution step by step.The solution to the system of matrix equations is as follows:

1. The matrix \( X \) is:
   \[
   X = \begin{pmatrix}
   0 & 1 \\
   0 & 3 \\
   0 & -4
   \end{pmatrix}
   \]

2. The matrix \( Y \) is:
   \[
   Y = \begin{pmatrix}
   2 & -1 \\
   0 & 0 \\
   2 & 2
   \end{pmatrix}
   \]

3. The product \( X^T \cdot Y \) (where \( X^T \) is the transpose of \( X \)) is:
   \[
   X^T \cdot Y = \begin{pmatrix}
   0 & 0 \\
   -6 & -9
   \end{pmatrix}
   \]

4. The determinant of \( X^T \cdot Y \) is:
   \[
   \text{det}(X^T \cdot Y) = 0
   \]

Would you like further details or clarifications?

Here are 5 related questions to expand the topic:
1. How do you find the transpose of a matrix?
2. What are the properties of matrix multiplication?
3. How is the determinant of a matrix calculated?
4. How do linear combinations of matrices work?
5. What happens when the determinant of a matrix is zero?

**Tip:** If the determinant of a matrix is zero, it means the matrix is singular and does not have an inverse.

Find the unknown matrices X and Y from the system of equations: -5X + 6Y = A and 4X - 5Y = B. Given matrices A and B, represent X and Y as linear combinations of matrices A and B, and calculate the product X^T * Y (where X^T is the transpose of matrix X). Finally, find the determinant of X^T * Y.

Learn how to solve matrix equations involving the unknown matrices X and Y, using linear combinations of matrices A and B. Calculate the product X^T * Y and determine the determinant of this product. Suitable for linear algebra 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