Determinants Flashcards
1
Q
When can a determinant be defined?
A
For a square matrix only (i.e. m = n).
2
Q
What happens when two rows/columns are interchanged?
A
The determinant is multiplied by -1.
3
Q
What happens when all values in a row or column are 0?
A
The determinant is 0.
4
Q
What happens when two columns or rows are multiples of each other?
A
The determinant = 0.
5
Q
det(A^T)
A
det(A)
6
Q
det(A*B)
A
det(A)*det(B)
7
Q
det(I)
A
1
8
Q
An m*n matrix has rank r > or = 1 iff:
A
- A has a at least one r*r sub-matrix with non-zero determinant.
- The determinant of every (r+1)*(r+1) sub-matrix is equal to 0.
9
Q
When does r = n (for a square matrix)?
A
Iff det(A) ~= 0. Therefore A*x = b has a unique solution iff det(A) ~= 0.
10
Q
How do you invert a matrix using determinants?
A
It is the transpose of adj(A) divided by det(A).
