Rank of a Matrix – Explained

The rank of a matrix is a fundamental concept in linear algebra. It represents the maximum number of linearly independent rows or columns in a matrix. In this document, we will explain the process of calculating the rank of a matrix.

Procedure

Let’s consider an $m \times n$ matrix $A$. To find the rank of matrix $A$, follow these steps:

1. Begin by writing down the given matrix $A$.
2. Apply row operations to bring the matrix into row-echelon form. This involves performing the following steps:

1. Start with the first row and locate the leftmost non-zero entry (pivot element). Swap rows if necessary to bring a non-zero entry to the top.
2. Perform row operations to make all the entries below the pivot zero. This can be achieved by subtracting a multiple of the first row from subsequent rows.
3. Move to the next row and repeat the above steps until you reach the last row or column.
4. The resulting matrix is in row-echelon form, with leading 1’s (pivots) in each row to the right of leading 1’s in the row above.
5. Count the number of non-zero rows in the row-echelon form of matrix $A$. This count is the rank of the matrix.

Example

Let’s illustrate the procedure with an example. Consider the following matrix:

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

Step 1: Writing the matrix

The given matrix is:

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

Step 2: Row operations and row-echelon form

Let’s perform row operations to bring the matrix into row-echelon form:

\[
\begin{bmatrix}
2 & 4 & 6 \\
1 & 3 & 5 \\
2 & 5 & 7
\end{bmatrix}
\xrightarrow{\text{R2 = R2 – 0.5R1}}
\begin{bmatrix}
2 & 4 & 6 \\
0 & 1 & 1 \\
2 & 5 & 7
\end{bmatrix}
\xrightarrow{\text{R3 = R3 – R1}}
\begin{bmatrix}
2 & 4 & 6 \\
0 & 1 & 1 \\
0 & 1 & 1
\end{bmatrix}
\]

\[
\xrightarrow{\text{R3 = R3 – R2}}
\begin{bmatrix}
2 & 4 & 6 \\
0 & 1 & 1 \\
0 & 0 & 0
\end{bmatrix}
\]

The resulting row-echelon form is:

\[
\begin{bmatrix}
2 & 4 & 6 \\
0 & 1 & 1 \\
0 & 0 & 0
\end{bmatrix}
\]

Step 3: Counting non-zero rows

In the row-echelon form, we have two non-zero rows: the first row and the second row. Therefore, the rank of matrix $A$ is 2.

Conclusion

The rank of a matrix is determined by bringing the matrix into row-echelon form and counting the number of non-zero rows. In this document, we demonstrated the process step-by-step using an example matrix. The rank of matrix $A$ was found to be 2.