Finding the rank through row operation

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In summary, to find the rank of a matrix, one can reduce the matrix to row-echelon form using elementary row operations and count the number of non-zero rows or columns. The rank is always equal to the smaller of these two numbers, as there will always be the same number or fewer non-zero rows than columns. Additionally, the rank of a matrix can also be determined by performing column reduction operations or row reduction operations, which will lead to the same result. The rank of a matrix essentially tells us the number of important (row or column) vectors in the matrix.
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timsea81
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Is the following statement correct?

To find the rank of a matrix, reduce the matrix using elementary row operations to row-echelon form. Count the number of not-all-zero rows and not-all-zero columns. The rank is smaller of those 2 numbers.
 
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timsea81 said:
Is the following statement correct?

To find the rank of a matrix, reduce the matrix using elementary row operations to row-echelon form. Count the number of not-all-zero rows and not-all-zero columns. The rank is smaller of those 2 numbers.

yes, but...

due to the nature of row-reduction, there will always be the same number or fewer non-zero rows than non-zero columns. every non-zero row has a "leading 1", which then also means the column containing that 1 (often called "pivot columns") is also non-zero.

but it may be that between two successive non-zero rows, the leading 1 in the lower row is more than 1 place to the right of the leading 1 in the upper row. since the entries in the upper row in the columns between the leading 1's are not constrained to be 0 (being neither above, nor below a leading 1), it can happen that they are, in fact, non-zero, leading to more non-zero columns than rows.

that's why it's called ROW reduction, because the rank of the original (and reduced row-echelon form) matrix is equal to the number of non-zero rows of the rref form.

one can also perform column reduction operations, leading to the number of non-zero columns being minimal. in this procedure, one obtains (perhaps) more non-zero rows than columns.

either way, the column rank of a column-reduced matrix, or the row rank of a row-reduced matrix, will give you the same number, which is simply called the rank of the original matrix. row rank is sometimes called "the dimension of the solution space", and column rank "the dimension of the image space", but in any case, one thing is clear: the rank of a matrix tells you essentially "how many (row, or column) vectors really matter".
 
  • #3
Thanks Deveno, that really helps clear things up.
 

Related to Finding the rank through row operation

What are row operations?

Row operations are a set of mathematical operations that can be applied to the rows of a matrix in order to transform it into a simpler or more useful form. These operations include swapping rows, multiplying a row by a non-zero constant, and adding a multiple of one row to another row.

Why is finding the rank of a matrix important?

The rank of a matrix is an important concept in linear algebra as it provides information about the number of linearly independent rows or columns in the matrix. This information can be used to determine the dimension of the vector space spanned by the rows or columns of the matrix, and can also be used to solve systems of linear equations.

What is the difference between a full rank and a reduced rank matrix?

A full rank matrix has a rank equal to the number of rows (or columns) in the matrix, meaning that all of its rows (or columns) are linearly independent. A reduced rank matrix has a rank that is less than the number of rows (or columns), meaning that some of its rows (or columns) are linearly dependent on others.

How do row operations help in finding the rank of a matrix?

Row operations can be used to transform a matrix into a simpler form, such as reduced row echelon form, which makes it easier to determine the rank. This is because the rank can be determined by counting the number of non-zero rows in the reduced matrix, which is equivalent to counting the number of linearly independent rows in the original matrix.

Can row operations change the rank of a matrix?

No, row operations do not change the rank of a matrix. They simply transform the matrix into a different form that is easier to work with and can provide more information about the matrix, but the rank remains the same.

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