Final answer:
The sum of two positive definite matrices may or may not be positive definite, depending on the sum matrix's eigenvalues. The set of all n by n positive matrices is not a vector space because it does not satisfy closure under scalar multiplication.
Step-by-step explanation:
In linear algebra, a positive definite matrix is a symmetric matrix that has all positive eigenvalues. When you add two positive definite matrices, the resulting matrix may or may not be positive definite. It depends on the matrices being added together. If the sum of the eigenvalues of the sum matrix is positive, then the sum matrix is positive definite.
No, the set of all n by n positive matrices is not a vector space. In order for a set to be a vector space, it must satisfy ten properties known as the vector space axioms. However, the set of all n by n positive matrices does not satisfy one of these axioms, which is closure under scalar multiplication. In other words, multiplying a positive matrix by a negative scalar can result in a matrix that is not positive.