Bases and Matrix Inverses

Question 1

Every consistent system has a unique solution ⇔ …

Answer 1

Columns of the coefficient matrix of size m × n are LI in R^m

Question 2

Every system is consistent ⇔ …

Answer 2

Columns of the coefficient

matrix span R^m

Question 3

Theorem 1
Suppose A is a n × n square matrix. Then the following
statements are equivalent: (11)

Answer 3

1. rank(A) = n,
2. rank(A^T) =n,
3. Every Ax = b has a unique solution,
4. RREF of A is In,
5. ker(A) = {0},
6. Col(A) = R^n,
7. Row(A) = R^n,
8. Columns of A are LI,
9. Rows of A are LI,
10. Columns of A form a
    basis for R^n,
11. Rows of A form a basis
    for R^n.

Question 4

Theorem 2

The following statements are equivalent for m × n matrix A: (6)

Answer 4

1. rank(A) = n,
2. Row(A) = R^n,
3. Columns of A: LI in R^m,
4. ker(A) = {0},
5. There exists a n × m matrix B such that BA = In,
6. The n × n matrix A^T A is
   invertible.

Question 5

Theorem 3

The following statements are equivalent for m × n matrix A: (6)

Answer 5

1. rank(A) = m,
2. Col(A) = R^m,
3. Rows of A: LI in R^n,
4. Ax = b is consistent for
   every b∈R^m.
5. There exists a n × m matrix B such that AB = Im,
6. The m × m matrix AA^T is
   invertible.

Question 6

Definition (Inverse of a matrix)

Answer 6

A n × n (square) matrix A is invertible if there is a n × n
matrix B such that
AB = In and BA = In,
where B is called the inverse of A (denoted by A^−1).
Inverses are unique (i.e. a matrix has at most one inverse)!

Question 7

How to find a 2x2 matrix’s inverse.

Answer 7

Suppose A =
[a b]
[c d]
and ad-bc≠0, A^-1 = 
[d -b]
[-c a] 1/(ad-bc)

Question 8

When is a 2x2 matrix not invertible?

Answer 8

When ad-bc=0

Question 9

Fact
Suppose A is an invertible n × n matrix. Then, any linear system
Ax = b: (2)

Answer 9

1. is consistent and,

2. has a unique solution.

Question 10

Algebraic Properties of Inverses
Fact
If k≠0 is a scalar and A, C are n × n invertible matrices, then: (5)

Answer 10

1. A^−1
and (A^−1)^−1 = A,
2. A^k
and (A^k)^−1 = (A^−1)^k,
3. A^T and (A^T)^−1 = (A^−1)^T,
4. kA and (kA)^−1 = (k^−1(A^−1),
5. AC and (AC)^−1 = (C^−1)(A^−1), (note the order)

are all invertible matrices. If AC is invertible, then so are A and C.

Question 11

What is the goal when finding matrix inverses?

Answer 11

To solve the equation AB=In for the unknown matrix B.

Question 12

What are 2 facts about invertible matrices?

Answer 12

1. A n × n matrix A is invertible ⇔ A is row equivalent to In,
2. If A is invertible, any sequence of elementary row operations
   that reduces A to In also transforms In to A^−1
   .

Question 13

Matrix Inversion Algorithm (4)

Answer 13

1. Form the superaugmented matrix [A|In],
2. Start row reducing to reduce A to its RREF,
3. If A is row equivalent to In, then [A|In] is row equivalent to [In|A^−1],
4. Otherwise, A is not invertible.

Question 14

What is a fact about invertible matrices and rank?

Answer 14

Suppose A is a n × n matrix with rank(A) = n. Then, A is invertible and A^−1
can be computed by the matrix inversion algorithm, or by row-reducing [A|In] to [In|A^−1].
If rank(A) < n, then A is not invertible.

Question 15

Big Theorem: Matrix Inverse Theorem
Let A be a n × n matrix. Then, the following statements are
equivalent: (17)

Answer 15

1. rank(A) = n,
2. Ax = 0 ⇒ x = 0,
3. Ax = b is consistent for
   every b∈R^n,
4. Every Ax = b has a unique solution.
5. RREF of A is In,
6. ker(A) = {0},
7. Col(A) = R^n,
8. Row(A) = R^n,
9. rank(A^T) = n,
10. Columns of A are LI,
11. Rows of A are LI,
12. Columns of A span R^n,
13. Rows of A span R^n,
14. Columns of A form a
    basis for R^n,
15. Rows of A form a basis
    for R^n,
16. A is invertible,
17. A^T is invertible.