Math Problem Statement

Find the normal form of the matrix A and hence find its rank, where \( A = \begin{bmatrix} 2 & 3 & -1 & -1 \\ 1 & -1 & -2 & -4 \\ 3 & 1 & 3 & -2 \\ 6 & 3 & 0 & -7 \end{bmatrix} \).

Solution

To solve this problem, we need to perform row operations to reduce the matrix AA to its normal form (Row Echelon Form or Reduced Row Echelon Form). From this, we can determine the rank of the matrix AA.

Given matrix: A=[23−1−11−1−2−4313−2630−7]A = \begin{bmatrix} 2 & 3 & -1 & -1 \\ 1 & -1 & -2 & -4 \\ 3 & 1 & 3 & -2 \\ 6 & 3 & 0 & -7 \end{bmatrix}

Step-by-Step Solution

  1. Make the leading entry of the first row 1 (if needed) and zero out the entries below it by applying row operations.
  2. Move to the next column and row, making that entry 1 and zeroing out entries below it.
  3. Repeat this process until we reach the last column or row possible.

Let me calculate and provide the row operations to find the normal form and rank.The matrix AA in its Row Echelon Form (also known as the normal form) is:

1 & 0 & 0 & -\frac{5}{3} \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & \frac{2}{3} \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}$$ From this form, we observe that there are three non-zero rows, which indicates that the **rank of the matrix $$A$$ is 3**. Would you like further details on the row operations used to reach this form? --- ### Related Questions 1. How can we interpret the rank of a matrix in terms of its linear independence? 2. What does it mean if a matrix has full rank? 3. How does the rank of a matrix affect the solutions to a system of linear equations? 4. Can a matrix have a rank higher than its smallest dimension (rows or columns)? 5. How can the rank be used to determine the invertibility of a square matrix? ### Tip For any matrix, the rank provides insight into the maximum number of linearly independent rows or columns.

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Row Echelon Form
Matrix Rank

Formulas

Row Echelon Form transformations
Rank of a matrix (number of non-zero rows in row echelon form)

Theorems

Rank theorem
Row Echelon Form properties

Suitable Grade Level

College level

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