Arizona State University (ASU) MAT343 Applied Linear Algebra Exam 2 Practice

When the rank of a matrix is equal to the number of its rows, it signifies that the matrix has full row rank. This means that all rows of the matrix are linearly independent. In linear algebra, the rank of a matrix is defined as the maximum number of linearly independent row or column vectors in the matrix. Therefore, if the rank equals the number of rows, it implies that none of the rows can be expressed as a linear combination of the others. This property is crucial because it often indicates that the corresponding system of linear equations, if represented by the matrix, has a unique solution. When you have full row rank and linear independence, it ensures that the information conveyed by each row is unique and not redundant. The other options do not accurately represent the implications of having the rank equal to the number of rows. For instance, linear dependence among rows (as suggested in another choice) would reduce the rank below the number of rows, thus contradicting the initial condition. A singular matrix typically has a rank less than the number of its rows, and claims about the presence of solutions to systems of equations would need additional context beyond just knowing the rank.