Final answer:
To prove that if K is any positive definite matrix, then every positive scalar multiple cK > 0 is also positive definite, we can use the definition of positive definite matrices.
Step-by-step explanation:
To prove that if K is any positive definite matrix, then every positive scalar multiple cK > 0 is also positive definite, we can use the definition of positive definite matrices. A matrix K is positive definite if for any non-zero vector X, X^KX > 0.
Now, let's consider a positive scalar multiple of K, denoted as cK. For any non-zero vector X, we have X^(cK)X = c(X^KX). Since K is positive definite, X^KX > 0. Since c is positive, c(X^KX) is also greater than 0.
Therefore, if K is any positive definite matrix, then every positive scalar multiple cK is also positive definite.