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The rank of a matrix is the number of lines in the matrix (rows or columns) that are linearly independent.

A line is linearly dependent on another one or others when a linear combination between them can be established.

A line is linearly independent of another one or others when a linear combination between them cannot be established.

The rank of a matrix is symbolized as rank(A) or r(A).

Calculating the Rank of a Matrix

The Gaussian elimination method is used to calculate the rank of a matrix.

A line can be discarded if:

1. All the coefficients are zeros.
2. There are two equal lines.
3. A line is proportional to another.
4. A line is a linear combination of others.

Since only two rows are unique that means the rank of matrix is

In general, eliminate the maximum possible number of lines, and the range is the number of nonzero rows.

Therefore

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Example

Calculate the rank of the following matrix:

Therefore, .

Calculating the rank of a matrix for determimants

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