https://r2.erweima.ai/mathbot/image/17248584491921972zy7w.jpg

resuelve las siguientes operaciones paso a paso

To solve the operations given in the matrices \( A \), I'll perform the calculations step by step for each one.

### Problem 4:
Given the matrix:
\[
A = \begin{pmatrix} -4 & 5 \\ 3 & -2 \end{pmatrix}
\]
For this matrix, I will assume the task is to calculate its determinant, which is a common operation associated with matrices.

The determinant of a \( 2 \times 2 \) matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[
\text{det}(A) = ad - bc
\]
Substituting the values from the matrix:
\[
\text{det}(A) = (-4)(-2) - (5)(3) = 8 - 15 = -7
\]
So, the determinant of matrix \( A \) is \(\boxed{-7}\).

### Problem 5:
Given the matrix:
\[
A = \begin{pmatrix} 6 & -2 \\ \frac{1}{4} & -\frac{1}{2} \end{pmatrix}
\]
Again, calculating the determinant of this matrix using the same formula:
\[
\text{det}(A) = 6\left(-\frac{1}{2}\right) - \left(-2\right)\left(\frac{1}{4}\right)
\]
Let's solve this step by step:
\[
\text{det}(A) = -3 + \frac{2}{4} = -3 + \frac{1}{2} = -\frac{6}{2} + \frac{1}{2} = -\frac{5}{2}
\]
So, the determinant of matrix \( A \) is \(\boxed{-\frac{5}{2}}\).

---

### Do you need any more details or have any questions?

Here are five related questions you might explore next:

1. How do you find the inverse of a \( 2 \times 2 \) matrix?
2. What are the eigenvalues and eigenvectors of the matrix in Problem 4?
3. How can you use determinants to solve systems of linear equations?
4. What is the geometric interpretation of the determinant of a matrix?
5. How does the determinant of a matrix change if you swap two rows or two columns?

**Tip:** Remember that the determinant of a matrix can give insight into the matrix's properties, such as whether it is invertible (non-zero determinant) or singular (zero determinant).

Explore detailed step-by-step solutions for calculating matrix determinants. Includes examples with matrices and their determinants, such as -7 and -5/2. Suitable for high school students and anyone interested in matrix algebra and determinants.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation