To find the normal form (also known as the row echelon form or reduced row echelon form) of the matrix $A$, and to determine its rank, we’ll use Gaussian elimination. Here’s the process step-by-step:

Matrix $A$

$A = \begin{bmatrix} 2 & 3 & -1 & -1 \\ 1 & -1 & -2 & -4 \\ 3 & 1 & 3 & -2 \\ 6 & 3 & 0 & -7 \end{bmatrix}$

Step 1: Create Leading 1 in the First Row, First Column

1. Divide the first row by 2: $R_1 \to \frac{1}{2} R_1 = \begin{bmatrix} 1 & \frac{3}{2} & -\frac{1}{2} & -\frac{1}{2} \end{bmatrix}$ So the matrix becomes: $\begin{bmatrix} 1 & \frac{3}{2} & -\frac{1}{2} & -\frac{1}{2} \\ 1 & -1 & -2 & -4 \\ 3 & 1 & 3 & -2 \\ 6 & 3 & 0 & -7 \end{bmatrix}$

2. Make the first element of rows 2, 3, and 4 zero by subtracting appropriate multiples of the first row from these rows.

Step 2: Proceed Through Row Operations for Row Echelon Form

Following Gaussian elimination, we’ll reach a matrix with a staircase form, after which we can identify the number of non-zero rows (pivots) to determine the rank.

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Further Questions:

1. What is the significance of a matrix's rank in linear algebra?
2. How do elementary row operations affect the determinant of a matrix?
3. What are the conditions for a matrix to be in reduced row echelon form?
4. How can rank be used to determine the consistency of a system of linear equations?
5. What are some applications of row echelon form in real-world problems?

Tip:

In Gaussian elimination, keeping track of each operation helps to avoid errors and verify each step's accuracy.