What are the properties of the sum of matrices and scalar multiplication? a) The sum is an internal composition law in Km x n. b) The sum is asso...

The answer is d) All of the above.

The sum of matrices and scalar multiplication are two important operations in matrix algebra. They have the following properties:

• The sum is an internal composition law in Km x n. This means that the sum of two matrices is another matrix, and the sum of a matrix and a scalar is another matrix.
• The sum is associative in Km x n. This means that the order in which we add matrices does not matter. For example, (A + B) + C = A + (B + C).
• The scalar multiplication is distributive over the sum of matrices. This means that we can distribute a scalar over the sum of two matrices. For example, k * (A + B) = k * A + k * B.

These properties are useful for simplifying expressions involving matrices and for proving other properties of matrices.

Here are some examples of how these properties can be used:

• To simplify the expression (A + B) + C, we can use the associative property of addition to get A + (B + C).
• To prove that the scalar multiplication is distributive over the sum of matrices, we can use the distributive property of real numbers.

These properties are essential for understanding and working with matrices.