Chapter 3 – Key Concepts

Question 1

What is the determinant of a matrix? (cofactor expansion on the first row)

Answer 1

For n ≥ 2, the determinant of an n×n matrix A = [a_ij] is the sum of n terms of the form ±a_1j det A_1j:

det A = a₁₁ det A₁₁ - … + (-1)ⁿ⁺¹ a_1n det A_1n

= ∑ (-1)^1+j a_1j det A_1j

Question 2

What is the cofactor expansion form of the determinant of A across the row i? What about down the column j?

Answer 2

det A = a_i1 C_i1 + … + a_in C_in

= a_1j C_1j + … + a_nj C_nj

where C_ij is the (i, j)-cofactor given by: C_ij = (-1)^i+j det A_ij

Question 3

If A is a triangular matrix, what product gives det A?

Answer 3

The product of the main diagonal entries gives det A.

Question 4

If A is a square matrix, determine the relationships between det A and det B:

a. If a multiple of one row of A is added to another row to produce a matrix B.
b. If two rows of A are interchanged to produce B.
c. If one row of A is multiplied by k to produce B.

Answer 4

a. det A = det B
b. det A = - det B
c. det B = k det A

Question 5

What is the determinant of an invertible matrix A in terms of its echelon form U? What about the determinant of a non-invertible matrix A?

Answer 5

For A invertible, det A = (-1)^r (product of pivots of U), where r is the number of row interchanges required to convert A to U.

For A not invertible, det A = 0

Question 6

If a square matrix A is invertible, what must be true about its determinant?

Answer 6

det A ≠ 0

Question 7

If A is an n×n matrix, what is the relationship between det A and det A^T?

Answer 7

det A^T = det A

Question 8

If A and B are n×n matrices, evaluate det AB.

Answer 8

det AB = (det A) (det B)

Question 9

What is Cramer’s rule?

Answer 9

Let A be an invertible n×n matrix. For any b ∈ ℝⁿ, the unique solution x of Ax = b has entries given by:

x_i = det A_i(b) / det A, i = 0, … n, where A_i(b) = [a₁ … b… a_n] where b is at column i.

Question 10

What is the inverse formula for a square matrix A. i.e. what is A^-1 in terms of adj A and det A?

Answer 10

A-1 = (1 / det A) adj A, where adj A = [[C₁₁ … C_n1] … [C_1n … C_nn] ] where []’s are rows of the matrix A.