lecture 4 - determinant part 2

Question 1

What is a submatrix?

Answer 1

A matrix obtained by certain cancelling rows/columns

Question 2

What is a minor?

Answer 2

Determinant of a submatrix

Question 3

What is a major?

Answer 3

Determinant of the largest square matrix

Question 4

What is a complement submatrix?

Answer 4

For element a(i,j), cancels out the ith row and jth column.

Question 5

What is complement minor?

Answer 5

Anotated by M(i,j)
Determinant of complement submatrix

Question 6

What is a cofactor?

Answer 6

A generic element of a complement minor.
Anoted A(i,j)
A(i,j) = (-1)^(i + j) * M(i,j)
(-1) alternated negation

Question 7

How to find adjugate matrix?

Answer 7

• transpose A > A^T
• find the complement minors (M(i,j)) for each element.
• find the cofactor for each element. (-1)^(i+j) * M(i,j)

Question 8

What is laplace theorem?

Answer 8

determinant of A can be computed as the scalar product of each row/column by the vector of its corresponding cofactor.
det(A) = SUM [ a(i,j) * A(i,j) for each i,j in a row.

Question 9

What is an inverse matrix?

Answer 9

For square matrix A, inverse matrix is A * A^-1 = 1 = I
I = identity matrix which is one diagonal line of 1s in a matrix of 0s.

Question 10

What is invertible?

Answer 10

If exists a square matrix B where AB = I = BA then matrix is said to be invertible.
Matrix is invertible IFF A is non-singular (det != 0)

Question 11

What is the invertible product?

Answer 11

(AB)^-1 = B^-1 * A^-1 where A and B are square and invertible

Question 12

When will a matrix have an inverse?

Answer 12

• if it is a square matrix
• AND if it has a determinant of NOT 0 (non-singular)

Question 13

how to calculate inverse of a matrix?

Answer 13

• find det(A)
• find the adjugate matrix
  > if 2 X 2, swap a,d + negate
  b,c
  > if 3 X 3 > use adj matrix formula (transpose, cofactor)
• multiply by 1 / det(A)