What does it mean for a matrix to be singular?

A matrix is described as singular primarily due to its determinant being equal to zero. This characteristic signifies that the matrix does not have an inverse. In linear algebra, an invertible matrix is critical for solving systems of equations, among other applications, as it allows for unique solutions. When the determinant is zero, it indicates that the rows (or columns) of the matrix are linearly dependent; that is, at least one row can be expressed as a linear combination of the others.

While the implications of having a zero determinant are significant, one way to visually interpret a singular matrix is to consider that it cannot span the entire space, leading to the situation where solutions to the associated linear system may not exist or may not be unique.

In contrast, the other options do not accurately describe the properties of a singular matrix. For example, a matrix having only one row does not inherently mean it is singular, as a non-zero single row matrix still has an inverse. Similarly, a matrix where all elements are equal to one may or may not be singular, depending on the size of the matrix. Lastly, while having a zero row or column could indicate a singular matrix, it is not a complete definition since not all singular matrices must exhibit this characteristic.