What is the rank of a matrix?

The rank of a matrix is defined as the dimension of its row space or the column space, which means it reflects the maximum number of linearly independent rows or columns in the matrix. This definition underscores the importance of the rank in understanding the linear relationships among the rows and columns. The rank indicates how many of the rows or columns are not redundant; that is, they cannot be written as a linear combination of the others.

For instance, if you have a matrix with a rank of 3, it means that there are 3 rows or 3 columns that are linearly independent, and all the other rows or columns can be expressed as combinations of these 3. This understanding is critical in linear algebra applications, particularly in solving systems of equations, determining the solutions' uniqueness, and analyzing vector spaces.

The other options provided do not accurately define the rank: the number of rows relates to the structure of the matrix but does not convey information about linear independence, while the sum of all entries gives a single numerical value without telling us about the matrix's dimensional properties. Recognizing the rank as a reflection of the dimension of the row or column space is essential in applied contexts like data analysis, optimization, and system modeling.