MAT 202 Midterm Question With Complete Solutions.

Rank Nullity Theorem Expanded for correct answer: dim(kerA) + dim(imA) = m rank(A) correct answer: dim(imA) nullity(A) correct answer: dim(kerA) Similarity between A and B correct answer: B = S-1AS or SB = AS Bases of R^N in Matrix correct answer: must be invertible subspace V of R^n dim(V) = m correct answer: can have at most m linearly independent vectors An Invertible Matrix can not have correct answer: a row of 0's Dimension of subspace V correct answer: is the number of vectors in a basis of V ker(A) = {0} means that correct answer: column vectors of A are linearly independent ker(A) correct answer: linear combination of the column vectors of A that equal 0 Subspace correct answer: must have 0 vector closed under addition and scalar multiplication Inverse of 2x2 Matrix correct answer: (1/det(A))[d -b] [-c a] (BA)^-1 correct answer: B^-1A^-1 BA = In correct answer: Both are invertible and A = B^-1 To Determine Invertibility and Inverse correct answer: Compute the rref([A|In]) A being original matrix In being identity solve the augmented matrix if you find [In|B] then it is invertible and B is inverse Matrix Multiplication is correct answer: Noncummutative Associative and Distributive