Hello, I have four 3x1 vector A, B, C, D, how can I get rank of them?

To determine the rank of the set of four 3x1 vectors \( A, B, C, \) and \( D \), you need to form a matrix with these vectors as columns and then compute the rank of that matrix. Here's a step-by-step method:

1. **Form the Matrix**: Create a 3x4 matrix where each column is one of the vectors \( A, B, C, \) and \( D \). Let's denote this matrix as \( M \).
   \[
   M = \begin{pmatrix}
   A_1 & B_1 & C_1 & D_1 \\
   A_2 & B_2 & C_2 & D_2 \\
   A_3 & B_3 & C_3 & D_3 \\
   \end{pmatrix}
   \]

2. **Row Reduce the Matrix**: Perform Gaussian elimination (or row reduction) on the matrix \( M \) to transform it into its row echelon form (REF) or reduced row echelon form (RREF).

3. **Count the Pivot Columns**: The rank of the matrix is equal to the number of leading 1's (pivots) in the row echelon form. Each pivot represents a linearly independent column vector in the original matrix.

Here is a more concrete example:

Let's say:
\[
A = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}, \quad
B = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}, \quad
C = \begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix}, \quad
D = \begin{pmatrix} d_1 \\ d_2 \\ d_3 \end{pmatrix}
\]

Then the matrix \( M \) will be:
\[
M = \begin{pmatrix}
a_1 & b_1 & c_1 & d_1 \\
a_2 & b_2 & c_2 & d_2 \\
a_3 & b_3 & c_3 & d_3 \\
\end{pmatrix}
\]

To find the rank of \( M \), follow these steps in more detail:

### Example Calculation
Suppose the vectors are:
\[
A = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad
B = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}, \quad
C = \begin{pmatrix} 7 \\ 8 \\ 9 \end{pmatrix}, \quad
D = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}
\]

1. Form the matrix:
\[
M = \begin{pmatrix}
1 & 4 & 7 & 1 \\
2 & 5 & 8 & 1 \\
3 & 6 & 9 & 1 \\
\end{pmatrix}
\]

2. Perform Gaussian elimination to find the REF or RREF:
\[
\begin{pmatrix}
1 & 4 & 7 & 1 \\
2 & 5 & 8 & 1 \\
3 & 6 & 9 & 1 \\
\end{pmatrix} \xrightarrow{R2 \leftarrow R2 - 2R1}
\begin{pmatrix}
1 & 4 & 7 & 1 \\
0 & -3 & -6 & -1 \\
3 & 6 & 9 & 1 \\
\end{pmatrix}
\xrightarrow{R3 \leftarrow R3 - 3R1}
\begin{pmatrix}
1 & 4 & 7 & 1 \\
0 & -3 & -6 & -1 \\
0 & -6 & -12 & -2 \\
\end{pmatrix}
\]

3. Continue the reduction:
\[
\begin{pmatrix}
1 & 4 & 7 & 1 \\
0 & -3 & -6 & -1 \\
0 & -6 & -12 & -2 \\
\end{pmatrix} \xrightarrow{R3 \leftarrow R3 - 2R2}
\begin{pmatrix}
1 & 4 & 7 & 1 \\
0 & -3 & -6 & -1 \\
0 & 0 & 0 & 0 \\
\end{pmatrix}
\]

4. The rank is the number of non-zero rows, which is 2 in this case.

Thus, the rank of the matrix formed by vectors \( A, B, C, \) and \( D \) is 2. This means that the maximum number of linearly independent vectors among \( A, B, C, \) and \( D \) is 2.

Would you like more details on any of these steps or have any questions?

Here are some related questions you might find interesting:
1. What are the properties of matrix rank?
2. How do you determine if vectors are linearly independent?
3. How can eigenvalues and eigenvectors be computed?
4. What is the significance of the determinant of a matrix?
5. How do you perform matrix operations such as addition and multiplication?

**Tip:** When checking for linear independence, always ensure the matrix has full rank to confirm that all vectors are linearly independent.

Learn how to calculate the rank of four 3x1 vectors A, B, C, and D using matrix operations. Step-by-step guide with examples and detailed explanation of Gaussian elimination method. Suitable for advanced high school students and anyone interested in linear algebra.

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