Reduce A to canonical form and find its rank if A=      1 2 3 0  2 4 3 2  3 2 1 3  6 8 7 5 

Given the matrix \( A \):

\[
A = \begin{bmatrix}
1 & 2 & 3 & 0 \\
2 & 4 & 3 & 2 \\
3 & 2 & 1 & 3 \\
6 & 8 & 7 & 5
\end{bmatrix}
\]

We will first reduce this matrix to its row echelon form (REF) and then to its reduced row echelon form (RREF). Afterward, we'll determine its rank by counting the number of non-zero rows.

### Step 1: Row Echelon Form (REF)

1. Start with the first pivot element in the first row, which is \( 1 \).
2. Use elementary row operations to make all entries below the pivot in the first column equal to zero.

For now, I will proceed to perform these operations step by step. Let me calculate that.After performing Gaussian elimination, the matrix \( A \) has been reduced to its row echelon form. The rank of the matrix \( A \) is \( 3 \), meaning there are three linearly independent rows.

Thus, the canonical form of the matrix has 3 non-zero rows, indicating that the rank of matrix \( A \) is \( \boxed{3} \).

Would you like to see the detailed steps of the row operations used to achieve this form, or do you have any other questions?

Here are 5 related questions to deepen understanding:

1. How do you find the null space of a matrix based on its canonical form?
2. Can the rank of a matrix change if we transpose it?
3. What is the connection between the rank of a matrix and its determinant?
4. How does the rank of a matrix relate to the solutions of a system of linear equations?
5. How can you tell if a matrix is invertible based on its rank?

**Tip**: A matrix's rank is equal to the number of pivots (leading ones) in its reduced row echelon form (RREF).

Reduce A to canonical form and find its rank if A= 1 2 3 0 2 4 3 2 3 2 1 3 6 8 7 5

Learn how to reduce the matrix A = [[1 2 3 0], [2 4 3 2], [3 2 1 3], [6 8 7 5]] to its canonical form and find its rank using Gaussian elimination. This problem covers key matrix theory concepts such as row echelon form, reduced row echelon form (RREF), and the rank theorem. Suitable for undergraduate students studying linear algebra.

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