Which property must be true for a matrix to be considered in a linear combination?

For a matrix to be considered in a linear combination, it is essential that it can be expressed using weights from scalar fields. This is because a linear combination involves multiplying matrices (or vectors) by scalars (these scalars are often referred to as weights) and then adding the resulting products. Scalar fields typically consist of real or complex numbers, and the ability to apply scalars to matrices allows for the construction of new matrices from existing ones while adhering to linearity.

In the context of linear combinations, any matrix can be represented as a sum of other matrices multiplied by appropriate scalar coefficients. This underpins many of the operations and properties explored in linear algebra, such as spanning sets, bases, and dimension.

Other choices relate to specific characteristics of matrices but do not capture the essence of what defines a linear combination. For instance, non-zero elements, square matrices, and dimensionality constraints might come into play in specific scenarios, but they are not requirements for defining a linear combination as per linear algebra principles. The critical aspect is the role of scalar multiplication and addition, which allows for the formation of linear combinations from matrices.