Determinants

Question 1

Suppose A is an nxn matrix, What is the determinant is n=1?

Answer 1

If n = 1, then A = [a] = a and det(A) = a

Question 2

Suppose A is an nxn matrix, What is the determinant is n=2?

Answer 2

If n = 2, then
det( 
[a b]
[c d])≡
|a b|
|c d|
= ad − bc,
Recall that a 2 × 2 matrix A is invertible ⇔ det(A)≠0.
If u = (a, b) and v = (c, d), then |det(A)| = ||u × v||.

Question 3

Suppose A is an nxn matrix, What is the determinant is n=3?

Answer 3

If n = 3, then
|a b c|
|d e f| =
|g h j|

a |e f| - b |d f| + c |d e|
|h j| |g j| |g h|

(This is called the cofactor expansion along the first row)

Question 4

Definition

Suppose A is a n×n matrix and 1 ≤ i, j ≤ n. Then, Aij is …

Answer 4

The (n − 1) × (n − 1) matrix obtained by removing the i-th row and j-th column of A.

Question 5

Define determinants.

Answer 5

For n ≥ 2, the determinant of a n × n matrix A = [aij] is
|A| = det(A) = a11det(A11) − a12det(A12) + · · ·
+(−1)^(1+n)a1ndet(A1n) …
(see notes)
which defines the determinant of A in terms of determinants of smaller matrices

Question 6

What’s a fact about cofactor expansion?

Answer 6

see notes

Question 7

What are two important points about cofactor expansion?

Answer 7

1. We can do the cofactor expansion along any row or column,
2. So we can choose rows or columns that make the calculation
   easy (e.g. rows or columns that have a lot of zeros).

Question 8

Properties of the Determinant

Let A be a n × n matrix. Then, … (2)

Answer 8

1. If A has a row or column of zeros, then det(A) = 0,

2. det(A) = det(A^T).

Question 9

Define triangular matrices.

Let A be a n × n matrix. Then, (3)

Answer 9

Let A be a n × n matrix. Then,
1. If A has all zeros below the main diagonal, it is called upper
triangular,
2. If A has all zeros above the main diagonal, it is called lower
triangular,
3. A is called triangular if it is upper or lower triangular.

Question 10

What is the Determinant of Triangular matrices?

Answer 10

If A is a triangular matrix, then det(A) is the product of the entries
of the main diagonal.

Question 11

Facts about the Determinant of Triangular matrices? (2)

Answer 11

If A is in RREF, then

1. rank(A) = n and det(A) =1,
2. rank(A) < n and det(A)=0.

Question 12

Row Operations and Determinants
Let A be a n × n matrix and suppose we do an elementary row
operation and obtain B. Then,
(3)

Answer 12

1. If B is obtained from A by adding a multiple of one row of A to another row, then det(A) = det(B),
2. If B is obtained from A by interchanging two rows of A, then det(B) = − det(A),
3. If B is obtained from A by multiplying a row of A by a scalar k, then det(B) = k det(A).
   These statements remain true, if we replace row by column.

Question 13

Row Operations and Determinants
Common Confusions
If we multiply a row by a scalar k, then (3)

Answer 13

If we multiply a row by a scalar k, then
1. det after the operation = k(det before the operation), or
2. det before the operation = 1/(k(det after the operation))
3. We only use column operations when computing determinants,
not when row reducing.

Question 14

More properties of determinants

Suppose A and B are n × n square matrices. Then (4)

Answer 14

Suppose A and B are n × n square matrices. Then,
1. det(rA) = (r^n)det(A), for any r∈R,
2. det(AB) = det(A) det(B),
3. det(A) = 0 ⇔ A is not invertible,
4. If A is invertible, then det(A^(−1)) = 1/det(A)
.