Midterm Exam

Question 1

What are all the chapter 1’s theorem’s?

Answer 1

Inverse are unique (aka there is only one inverse for a matrix)
(A^-1)^-1 = A aka A inverse inverse equals to A
(AB)^-1 = B^-1A^-1
(A^T)^-1 = (A^-1)^T (T is transpose here)
For some constant f, (fA)^-1 = (1/f)(A^-1)
If a matrix A is invertible than the matrix equation Ax = b has a unique solutions x = A^-1*b

Question 2

What are all the chapter 1’s definition?

Answer 2

Nonsingular: Has an inverse
Singular: Does not have an inverse

Question 3

How do you solve problems where you need to prove a matrix A is invertible?

Answer 3

Look for a different matrix or set of matrices such that A times it is equal to the identity matrix. Thus, by definition, it must the matrix must be its inverse.