How do you compute the determinant of an nxn matrix A, n>=2?
det A = a11detA11 - a12detA12 + … + (-1)^1+n * a1ndetA1n
What is the (i,j)-cofactor of A?
Cij = (-1)^i+j * detAij
How can the determinant be computed (cofactor)?
cofactor expansion:
- expansion across ith row : det A = ai1Ci1 + ai2Ci2 + … + ainCin
- expansion across jth column : det A = a1jC1j + a2jC2j + … + anjCnj
What is the determinant of a triangular matrix?
det A = product of entries on main diagonal
If A and B are square, what are the properties of the determinant?
- if a multiple of one row of A is added to another row to produce a matrix C : det A = det C
- if two rows of A are interchanged to produce C : det C = -det A
- if one row of A is multiplied by k to produce C : det C = kdet A
- A invertible <=> det A /=0
- det A^T = det A
- det AB = (det A) (det B)
What is Ai(b)?
the matrix from A by replacing column i by b
What is Cramer’s rule?
A nxn and invertible : the unique solution x of Ax=b has entries
xi = (det Ai(b))/det A
What is the adjugate of A adj A?
matrix with entries Cij
If A is nxn and invertible, what is A^-1?
A^-1 = (1/det A) * adj A
If A is 2x2, what is the absolute value of det A (/det A/)?
the area of the parallelogram determined by columns of A
What is the volume of the parallelepiped determined by the columns of a 3x3 matrix A?
absolute value of det A (/det A/)
