Chapter 3

Question 1

What is the determinant of a matrix A?

Answer 1

Let A =[a_ij] be an n x n matrix. If n = 1, then the determinant of A, denoted det A, is the single entry of A. For n > 1,
det A = a_11 det A_11 - a_12 Det A_12 + … +
(-1)^(n+1) a_1n det A_1n
where A_ij is the submatrix formed by ommitting the ith row and the jth column of A

Question 2

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

Answer 2

Let A be an n x n matrix. The (i,j) - cofactor of A is the signed determinant
C_ij = (-1)^(i+j) det A_ij

Question 3

How can the determinant of a matrix A be computed in general? (Theorem 3.1)

Answer 3

The determinant of the n x n matrix A can be computed by a cofactor expansion across any row or down any column of A
Cofactor expansion along the ith row yields
det A = ai1 Ci1 + ai2 Ci2 + … + ain Cin
Cofactor expansion along the jth column yields
det A = a1j C1j + a2j C2j + … + anj Cnj

Question 4

How is the determinant computed for a triangular matrix? (Theorem 3.2)

Answer 4

If A is a triangular matrix (upper or lower), the det A is the product of the entries on the main diagonal.

Question 5

What are the effects of elementary operations on the determinant of A? (Theorem 3.3)

Answer 5

If A is an n x n matrix and E is an n x n elementary matrix, then
det EA = det E * det A. So, if B = EA, then:
a) If E represents a row replacement, then
det B = det A
b) If E represents an interchange, then
det B = - det A
c) If E represents scaling a row by k, then
det B = k det A

Question 6

Theorem 3.4

Answer 6

An n x n matrix A is invertible if and only if det A is not equal to 0

Question 7

Theorem Properties of the determinant

Answer 7

If A and B are n x n matrices, then

i) det A^T = det A
ii) det(AB) = det A * det B
iii) det A^-1 = 1/(det A) when det A is not equal to zero

Question 8

Cramer’s rule

Answer 8

Let A be an invertible n x n matrix. For any b in R^n, the unique solution x of Ax = b has entries given by:
xi = ( det Ai (b) ) / (det A)
where Ai(b) is the matrix formed when you replace the ith column of the A matrix with the entries of b.