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Rank of a Matrix – What is it? How to find it?

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The rank of a matrix is the number of independent rows of a matrix. The rank of a matrix can be determined easily by converting the matrix into row echelon form and counting the number of nonzero rows.

The matrix can be converting into row echelon form by means of elementary row operations. This involves multiplying one row by a number and subtracting it from another row. Swapping two rows is also allowed. The matrix is bought in row echelon form when each row has a leading nonzero number (called the pivot) and the numbers below every pivot element are zero. The pivot of a succeeding row should come after the pivot of the preceding row. Let us try to understand this by looking at some examples of calculating the rank of a matrix.

Example 1:

Find the rank of the following matrix,

\begin{equation*} \begin{bmatrix} 1 & 5 & 1 \\ 2 & 5 & 10 \\ 3 & 15 & 2 \end{bmatrix} \end{equation*}

Solution:

We convert the matrix in row echelon form as follows,

Finding rank of a matrix by converting it to echelon form

Since the above matrix in echelon form has two nonzero rows, we conclude that the rank of the original matrix is 2.

Example 2:

Determine the rank of the following matrix,

\begin{equation*} \begin{bmatrix} 1 & 3 & 2 & 4 & 3\\ 3 & 2 & 4 & 7 & 6\\ 5 & 8 & 8 & 8 & 5 \\ 7 & 6 & 9 & 9 & 2\\ \end{bmatrix} \end{equation*}

Solution:

We convert the matrix in row echelon form as follows,

Finding rank of a matrix by converting it to echelon form
Finding rank of a matrix by converting it to echelon form
Example of finding rank of a matrix by converting it to echelon form

Since the above matrix has four nonzero rows, we conclude that the rank of the original matrix is 4.

Since the matrix has the maximum possible rank, such a matrix is called a full rank matrix.

Example 3:

Calculate the rank of the following matrix,

\begin{equation*} \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 7 & 5 \end{bmatrix} \end{equation*}

Solution:

We convert the matrix in row echelon form as follows,

Example of finding rank of a matrix by converting it to echelon form

Since the above matrix has two nonzero rows, we conclude that the rank of the original matrix is 2.

Remarks:

1. The rank of a matrix is always less or equal to both the total number of rows and the total number of columns.

2. We can also find the rank by using the determinant method. It involves finding the smallest size of the minor whose determinant is nonzero.

3. The number of independent columns of a matrix is equal to the number of independent rows. This implies that the rank of a matrix is equal to the rank of its transpose.

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