Lecture Chapter 3

Question 1

How do you compute the determinant of an nxn matrix A, n>=2?

Answer 1

det A = a11detA11 - a12detA12 + … + (-1)^1+n * a1ndetA1n

Question 2

What is the (i,j)-cofactor of A?

Answer 2

Cij = (-1)^i+j * detAij

Question 3

How can the determinant be computed (cofactor)?

Answer 3

cofactor expansion:

• expansion across ith row : det A = ai1Ci1 + ai2Ci2 + … + ainCin
• expansion across jth column : det A = a1jC1j + a2jC2j + … + anjCnj

Question 4

What is the determinant of a triangular matrix?

Answer 4

det A = product of entries on main diagonal

Question 5

If A and B are square, what are the properties of the determinant?

Answer 5

• 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)

Question 6

What is Ai(b)?

Answer 6

the matrix from A by replacing column i by b

Question 7

What is Cramer’s rule?

Answer 7

A nxn and invertible : the unique solution x of Ax=b has entries
xi = (det Ai(b))/det A

Question 8

What is the adjugate of A adj A?

Answer 8

matrix with entries Cij

Question 9

If A is nxn and invertible, what is A^-1?

Answer 9

A^-1 = (1/det A) * adj A

Question 10

If A is 2x2, what is the absolute value of det A (/det A/)?

Answer 10

the area of the parallelogram determined by columns of A

Question 11

What is the volume of the parallelepiped determined by the columns of a 3x3 matrix A?

Answer 11

absolute value of det A (/det A/)