https://file.aiquickdraw.com/mathbot/file/1726164630115t4oiykdm.jpg

To find the rank of matrix \( A \), let's start by writing out the matrix \( A \):

\[
A = \begin{bmatrix}
x - 1 & x + 1 & x \\
-1 & x & 0 \\
0 & 1 & 1
\end{bmatrix}
\]

### Step 1: Row-reduce the matrix to its echelon form.
We will perform row operations to simplify the matrix.

- Start with the first row:
  \[
  R_1 = [x - 1, x + 1, x]
  \]
  
- Subtract \( \frac{-1}{x - 1} \) times \( R_1 \) from \( R_2 \) (if \( x \neq 1 \)) to make the first entry of the second row 0.

- Then, continue similarly for the third row to get the matrix in echelon form.

### Step 2: Determine the rank.
Once the matrix is in echelon form, the rank will be equal to the number of non-zero rows. This depends on the value of \( x \), which can result in different scenarios where the rank of the matrix changes.

#### Special Cases:
- For some values of \( x \), such as \( x = 1 \), certain rows might become dependent (linearly dependent), affecting the rank.

Let me proceed with the detailed row reduction calculation to determine the general and special cases for the rank of \( A \).The row-reduced echelon form of the matrix \( A \) is:

\[
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\]

This means that for all real values of \( x \), the matrix \( A \) reduces to the identity matrix, which has three pivot columns. Therefore, the **rank of matrix \( A \) is 3**.

Would you like any additional details or further explanation?

Here are 5 related questions to expand your understanding:

1. What is the significance of the rank of a matrix in linear algebra?
2. How does the row-reduction method work to determine matrix rank?
3. Can the rank of a matrix change if the values in the matrix are changed slightly?
4. What happens to the rank if a row or column becomes linearly dependent?
5. How does the determinant of a matrix relate to its rank?

**Tip:** The rank of a matrix tells you the number of linearly independent rows or columns, which is useful for understanding the solutions to systems of linear equations.

Find the rank of the matrix A, where A is a 3x3 matrix given by: A = [[x - 1, x + 1, x], [-1, x, 0], [0, 1, 1]], and x is a real number.

Learn how to find the rank of a 3x3 matrix A with variable x, using row reduction techniques to bring the matrix to its echelon form. Understand how the rank is determined by the number of pivot columns and explore different scenarios based on the value of x. Suitable for undergraduate students studying Linear Algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation