Matrices T2

Question 1

what is an upper triangular matrix

Answer 1

a matrix with all non zero entries above or on the main diagonal

Question 2

what are matrix determinant properties?

Answer 2

• if there’s a row or column of all zeroes the det A = 0
• If a row of A is a multiple of some other row of A, then det A = 0. Similar for columns.
• if A is an upper triangular matrix the determinant is the product along the diagonal

Question 3

what are the elementary row operations

Answer 3

• add one row to another
• multiply a row by a non zero scalar
  -add multiples of a row to another row
• swapping rows

Question 4

what does it mean for a matrix too be in reduced row echelon form

Answer 4

A matrix in REF is further said to be in reduced row ech- elon form (RREF), if all leading entries are 1 and if each leading entry is the only non-zero entry in its column.

Question 5

what is the rank of a matrix

Answer 5

The rank of a matrix is the number of non-zero rows in its REF.

Question 6

if the REF of the augmented matrix has a row of zeros on the A-side, but the corresponding RHS entry is different from zero, then the system has no solutions

Answer 6

if the REF of the augmented matrix has a row of zeros on the A-side, but the corresponding RHS entry is different from zero, then the system has no solutions

Question 7

what is an eigenvalue and an eigenvector

Answer 7

EigenvaluesandEigenvectors. Let A be a n×n square matrix. A scalar λ ∈ R is called an eigenvalue of A, if there exists a n-vector v ̸= 0 with
A⃗v = λ⃗v.
Such a vector ⃗v is called an eigenvector of A.