Linear algebra

Integer linear programming: formulations, techniques and applications.

03

Santiago Valdés Ravelo [email protected] August, 2020 “Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.”1

Michael F. Atiyah.

1Michael F. Atiyah, “Special Article: Mathematics in the 20푡ℎ Century”, Bulletin of the London Mathematical Society, 2002, p. 7. 1 Vectors

2 Matrices

3 Linear systems

4 Exercises Vectors Vectors

Definition

Finite sequence of real values:

⎫ value 1 } } } value 2 } } } value 3 } 푛 + ⎬ 푣 ∈ ℝ , where 푛 ∈ ℕ … } } value 푛 − 1 } } } value 푛 } } ⎭ Vectors

Special vectors. Null

Vector with all the elements equal to zero (null):

⎫ 0 } } 0 } } 푣 = [0, 0, … , 0]푡 } … 푣 ⎬ } 0 } } } 0 } ⎭ Vectors

Special vectors. Unit

Vector with one element equals to 1 and the rest equal to 0:

⎫ 0 } } } … } } 0 } } 푣 = [0, … , 0, 1, 0, … , 0]푡 } 1 ⎬ 푣 } 0 } } } … } } 0 } } * unit vectors are those whose norm is equals to 1 ⎭ Vectors

Operations. Sum

Sum the elements at the same position:

⎫ } 푛 푣1 + 푤1 } 푣, 푤 ∈ ℝ ∶ } } 푣2 + 푤2 } 푣 + 푤 = [푣 , 푣 , … , 푣 ]푡 + [푤 , 푤 , … , 푤 ]푡 푣 + 푤 1 2 푛 1 2 푛 … ⎬ } } 푡 푣푛 + 푤푛 } 푣 + 푤 = [푣1 + 푤1, 푣2 + 푤2, … , 푣푛 + 푤푛] } } ⎭ Vectors

Operations. Scalar multiplication

Multiply each element of the vector by the scalar (real number):

⎫ } 푛 휆 × 푣1 } 휆 ∈ ℝ, 푣 ∈ ℝ ∶ } } 휆 × 푣2 } 휆 × 푣 = 휆 × [푣 , 푣 , … , 푣 ]푡 휆 × 푣 1 2 푛 … ⎬ } } 푡 휆 × 푣푛 } 휆 × 푣 = [휆 × 푣1, 휆 × 푣2, … , 휆 × 푣푛] } } ⎭ Vectors

Operations. Internal product

Multiply the elements at the same position and sum the multiplication results:

푣, 푤 ∈ ℝ푛 ∶

푤1 ⎡ 푤 ⎤ 푣푡 × 푤 = [푣 , 푣 , … , 푣 ] × ⎢ 2 ⎥ 1 2 푛 ⎢ … ⎥

⎣ 푤푛 ⎦ 푡 푛 푣 × 푤 = ∑푖=1 (푣푖 × 푤푖)

* the result of the internal product is a scalar. Vectors

Linear combination

Given 푚 vectors:

푣1, 푣2, … , 푣푚 ∈ ℝ푛. A of them is:

푚 1 2 푚 푖 휆1 × 푣 + 휆2 × 푣 + … + 휆푚 × 푣 = ∑ (휆푖 × 푣 ). 푖=1

Where, 휆1, 휆2, …, 휆푚 are scalars:

휆1, 휆2, … , 휆푚 ∈ ℝ. Vectors

Linear combination

1 2 푚 푚 푖 푣1 푣1 푣1 ∑푖=1 휆푖 × 푣1

1 2 푚 푚 푖 푣2 푣2 푣2 ∑푖=1 휆푖 × 푣2 휆1× + 휆2× + … + 휆푚× = … … … …

1 2 푚 푚 푖 푣푛 푣푛 푣푛 ∑푖=1 휆푖 × 푣푛 Vectors

Affine combination

If the sum of the scalars in a linear combination is equal to 1, then the combination is an AFFINE combination.

Affine combinations are linear combinations where:

푚

∑ 휆푖 = 1. 푖=1 Vectors

Conical combination

If the the scalars in a linear combination are non-negative, then the combination is a CONICAL combination.

Conical combinations are linear combinations where:

휆푖 ∈ ℝ+, ∀푖 ∈ {1, 2, … , 푚}. Vectors

Convex combination

If a linear combination is affine and conical, then the combination is a .

Convex combinations are linear combinations where:

푚

∑ 휆푖 = 1 푖=1 휆푖 ∈ ℝ+, ∀푖 ∈ {1, 2, … , 푚}. Vectors

Linear subspace

푆 ⊆ ℝ푛 is a if for any two vectors 푣, 푤 ∈ 푆 and any two scalars 훼, 훽 ∈ ℝ, the vector resulting from the linear combination 훼 × 푣 + 훽 × 푤 is in 푆. Vectors

Affine subspace

푆 ⊆ ℝ푛 is an AFFINE SUBSPACE if for any two vectors 푣, 푤 ∈ 푆 and any two scalars 훼, 훽 ∈ ℝ, where 훼 + 훽 = 1, the vector resulting from the affine combination 훼 × 푣 + 훽 × 푤 is in 푆.

* Every linear subspace is an affine subspace. However, the opposite does not hold true. Vectors

Conical set

푆 ⊆ ℝ푛 is a CONICAL SET if for any two vectors 푣, 푤 ∈ 푆 and any two non-negative scalars 훼, 훽 ∈ ℝ+, the vector resulting from the conical combination 훼 × 푣 + 훽 × 푤 is in 푆.

* Every linear subspace is a conical set. However, the opposite does not hold true. Vectors

Convex set

푆 ⊆ ℝ푛 is a if for any two vectors 푣, 푤 ∈ 푆 and any two non-negative scalars 훼, 훽 ∈ ℝ+, where 훼 + 훽 = 1, the vector resulting from the convex combination 훼 × 푣 + 훽 × 푤 is in 푆.

* Every conical, affine or linear subspace is a convex set. However, the opposite does not holdtrue. Vectors

Linear independence

The vectors 푣1, 푣2, … , 푣푚 ∈ ℝ푛 are LINEARLY INDEPENDENT if there exists only one linear combination resulting in the null vector, and is when all the scalars of the combination are equal to zero, i.e.:

푚 푖 푡 ∑ (휆푖 × 푣 ) = [0, 0, … , 0] ⇒ 휆푖 = 0, ∀푖 ∈ {1, 2, … , 푚} 푖=1 Vectors

Affine independence

The vectors 푣1, 푣2, … , 푣푚 ∈ ℝ푛 are AFFINELY INDEPENDENT if:

푚 푖 푡 ∑푖=1 (휆푖 × 푣 ) = [0, 0, … , 0] ⇒ 휆푖 = 0, ∀푖 ∈ {1, 2, … , 푚} 푚 ∑푖=1 휆푖 = 0

* Any set of linearly independent vectors is affine independent. However, the opposite does not hold true. Vectors

Linear span (hull)

The (also called LINEAR HULL) of a set 푆 ⊆ ℝ푛 is the subspace containing all the linear combinations of the elements of 푆:

푚 푖 푖 푠푝푎푛(푆) = {∑ (휆푖 × 푣 ) | 푚 ∈ ℕ, 푣 ∈ 푆, 휆푖 ∈ ℝ} 푖=1 Vectors

Affine span (hull)

The AFFINE SPAN (also called ) of a set 푆 ⊆ ℝ푛 is the subspace containing all the affine combinations of the elements of 푆:

푚 푚 푖 푖 푎푓푓(푆) = {∑ (휆푖 × 푣 ) | 푚 ∈ ℕ, 푣 ∈ 푆, 휆푖 ∈ ℝ, ∑ 휆푖 = 1} 푖=1 푖=0 Vectors

Conical hull

The CONICAL HULL of a set 푆 ⊆ ℝ푛 is the set containing all the conical combinations of the elements of 푆:

푚 푖 푖 푐표푛푒(푆) = {∑ (휆푖 × 푣 ) | 푚 ∈ ℕ, 푣 ∈ 푆, 휆푖 ∈ ℝ+} 푖=1 Vectors

Convex hull

The of a set 푆 ⊆ ℝ푛 is the set containing all the convex combinations of the elements of 푆:

푚 푚 푖 푖 푐표푛푣푒푥(푆) = {∑ (휆푖 × 푣 ) | 푚 ∈ ℕ, 푣 ∈ 푆, 휆푖 ∈ ℝ+, ∑ 휆푖 = 1} 푖=1 푖=0 Vectors

Basis

A set 푆 ⊂ ℝ푛 is a BASIS of ℝ푛, if the elements of 푆 are linearly independent and the linear span of 푆 is ℝ푛.

Remark. If 푆 is a basis of ℝ푛, then for any vector 푣 ∈ ℝ푛, there exists exactly one linear combination of the vectors of 푆 equals to 푣. Vectors

Change of basis

Consider a basis 푆 of ℝ푛 and a vector 푣 ∈ ℝ푛.

If the vector 푤 ∈ 푆 is multiplied by a scalar 휆 ≠ 0 in the linear combination of the elements of 푆 that results in 푣, then (푆 ⧵ {푤}) ∪ {푣} is a basis of ℝ푛. Matrices Matrices

Definition

Rectangular array of real values:

⎫ value 1, 1 value 1, 2 … value 1, 푛 } } } value 2, 1 value 2, 2 … value 2, 푛 } 푚×푛 + ⎬ 퐴푚×푛 ∈ ℝ ,(푚, 푛 ∈ ℕ ) … … … … } } value 푚, 1 value 푚, 2 … value 푚, 푛 } } ⎭ Matrices

Rows

푛 푎푖. ∈ ℝ is the 푖-th row of of the matrix 퐴푚×푛 (1 ≤ 푖 ≤ 푚):

value 1, 1 value 1, 2 … value 1, 푛

… … … …

value 푖, 1 value 푖, 2 … value 푖, 푛 푎푖. →

… … … …

value 푚, 1 value 푚, 2 … value 푚, 푛 Matrices

Columns

푚 푎.푗 ∈ ℝ is the 푗-th column of the matrix 퐴푚×푛 (1 ≤ 푗 ≤ 푛):

푎.푗 ↓

value 1, 1 … value 1, 푗 … value 1, 푛

value 2, 1 … value 2, 푗 … value 2, 푛

… … … … …

value 푚, 1 … value 푚, 푗 … value 푚, 푛 Matrices

Special matrices. Null

Each column of the matrix is a null vector (i.e. all elements are equal to zero):

0 … 0 … 0

… … … … …

0 … 0 … 0

… … … … …

0 … 0 … 0 Matrices

Special matrices. Identity

Square matrix (푚 = 푛), where 푎.푗 is the unit vector where the value 1 occurs at position 푗 (1 ≤ 푗 ≤ 푛) (i.e. all elements are equal to zero except for the main diagonal, whose elements are equal to one):

⎫ 1 0 0 … 0 } } 0 1 0 … 0 } } } 0 0 ⋱ ⋱ … 퐼 ∈ ℝ푛×푛 ⎬ 푛×푛 ⋱ ⋱ } … … 0 } } } 0 0 … 0 1 } ⎭ Matrices

Special matrices. Lower triangular

A square matrix is LOWER TRIANGULAR if all the values above the main diagonal are equal to zero (i.e. if 푖 < 푗, then 푎푖,푗 = 0):

푎1,1 0 0 … 0

푎2,1 푎2,2 0 … 0

⋱ 푎3,1 푎3,2 푎3,3 …

… … ⋱ ⋱ 0

푎푛,1 푎푛,2 … 푎푛,푛−1 푎푛,푛 Matrices

Special matrices. Upper triangular

A square matrix is UPPER TRIANGULAR if all the values below the main diagonal are equal to zero (i.e. if 푖 > 푗, then 푎푖,푗 = 0):

푎1,1 푎1,2 푎1,3 … 푎1,푛

0 푎2,2 푎2,3 … 푎2,푛

⋱ 0 0 푎3,3 …

⋱ ⋱ … … 푎푛−1,푛

0 0 … 0 푎푛,푛 Matrices

Operations. Scalar multiplication

Multiply each element of the matrix by the scalar:

⎫ } 휆 × 푎1,1 휆 × 푎1,2 … 휆 × 푎1,푛 } 푚×푛 } 휆 ∈ ℝ, 퐴푚×푛 ∈ ℝ ∶ } 휆 × 푎2,1 휆 × 푎2,2 … 휆 × 푎2,푛 } 휆 × 퐴 ∈ ℝ푚×푛 … … … … ⎬ } } 휆 × 푎푚,1휆 × 푎푚,2 … 휆×푎푚,푛 } } } ⎭ Matrices

Operations. Scalar sum

Sum each element of the matrix with the scalar:

⎫ } 휆 + 푎1,1 휆 + 푎1,2 … 휆 + 푎1,푛 } 푚×푛 } 휆 ∈ ℝ, 퐴푚×푛 ∈ ℝ ∶ } 휆 + 푎2,1 휆 + 푎2,2 … 휆 + 푎2,푛 } 휆 + 퐴 ∈ ℝ푚×푛 … … … … ⎬ } } 휆 + 푎푚,1휆 + 푎푚,2 … 휆+푎푚,푛 } } } ⎭ Matrices

Operations. Sum

Sum the elements at the same position (row and column):

⎫ } 푎1,1 + 푏1,1 푎1,2 + 푏1,2 … 푎1,푛 + 푏1,푛 } 푚×푛 }퐴푚×푛, 퐵푚×푛 ∈ ℝ ∶ } 푎2,1 + 푏2,1 푎2,2 + 푏2,2 … 푎2,푛 + 푏2,푛 } 퐴 + 퐵 ∈ ℝ푚×푛 … … … … ⎬ } } 푎푚,1 + 푏푚,1 푎푚,2 + 푏푚,2 … 푎푚,푛 + 푏푚,푛 } } } ⎭ Matrices

Operations. Multiplication

푚×푛 The matrix multiplication 퐴 × 퐵, where 퐴푚×푛 ∈ ℝ and 푝×푞 퐵푝×푞 ∈ ℝ , can be calculated iff 푛 = 푝 (i.e. the number of columns of the first matrix must be equal to the number of linesof the second one).

The resulting matrix 퐶 = 퐴 × 퐵 satisfies: 퐶 ∈ ℝ푚×푞 and the element 푐푖푗 is equal to the internal product of row 푎푖. of 퐴 by column 푏.푗 of 퐵 (1 ≤ 푖 ≤ 푚 and 1 ≤ 푗 ≤ 푞).

* Matrix multiplication is not a symmetric operation. Matrices

Operations. Multiplication

푚×푛 푛×푞 퐴푚×푛 ∈ ℝ , 퐵푛×푞 ∈ ℝ ∶

⎫ ∑푛 푎 × 푏 ∑푛 푎 × 푏 ∑푛 푎 × 푏 } 푖=1 1,푖 푖,1 푖=1 1,푖 푖,2 … 푖=1 1,푖 푖,푞 } } ∑푛 푎 × 푏 ∑푛 푎 × 푏 ∑푛 푎 × 푏 } 푖=1 2,푖 푖,1 푖=1 2,푖 푖,2 … 푖=1 2,푖 푖,푞 } 퐴 × 퐵 ∈ ℝ푚×푞 … … … … ⎬ } } ∑푛 푎 × 푏 ∑푛 푎 × 푏 ∑푛 푎 × 푏 푖=1 푚,푖 푖,1 푖=1 푚,푖 푖,2 … 푖=1 푚,푖 푖,푞 } } } ⎭ Matrices

Operations. Transpose

Given the matrix 퐴 ∈ ℝ푚×푛, the transpose is 퐴푡 ∈ ℝ푛×푚, where 푡 푡 the column 푎.푗 of 퐴 is equal to the row 푎푗. of 퐴 (1 ≤ 푗 ≤ 푛).

If 퐴 = 퐴푡 then 퐴 is SYMMETRIC and if 퐴 = −퐴푡 then 퐴 is ANTISYMMETRIC. Matrices

Inverse

A square matrix 퐴푛×푛 is INVERTIBLE (also known as non-singular or non-degenerate) if there exists a matrix 퐵푛×푛 (INVERSE), such that the product of 퐴 and 퐵 is equal to the identity:

퐴 × 퐵 = 퐼푛×푛

The INVERSE of 퐴 is denoted by 퐴−1.

* 퐴푛×푛 is invertible iff the lines of 퐴 are linearly independent. Matrices

Determinant

푛×푛 Each square matrix 퐴푛×푛 ∈ ℝ has associated a scalar number named DETERMINANT (푑푒푡(퐴) or |퐴|).

푛 푑푒푡(퐴) = |퐴| = ∑ 푠푖푔푛(휋) ∏ 푎 . 푖휋푖 휋∈풫푛 푖=1

Where, 풫푛 is the set of all permutations of the elements in {1, … , 푛} and 푠푖푔푛(휋) is equal to (−1)푘, being 푘 the number of two elements swap required to reorder 휋.

* The determinant of a 푛 × 푛 matrix can be computed within 풪 (푛2.373) complexity time. Matrices

Rank

푚×푛 The RANK of a matrix 퐴푚×푛 ∈ ℝ is an integer value equals to the maximum number of matrix rows that are linearly independent.

푟푎푛푘(퐴) = max |퐿|. 퐿 ⊆ {푎푖.|1 ≤ 푖 ≤ 푛} and 퐿 is linearly independent

* if 퐴 is a 푛 × 푛 matrix and 푑푒푡(퐴) ≠ 0, then 푟푎푛푘(퐴) = 푛. Linear systems Linear systems

Definition

푚×푛 A LINEAR SYSTEM receives a matrix 퐴푚×푛 ∈ ℝ and a vector 푏 ∈ ℝ푚, and seeks for a vector 푥 ∈ ℝ푛, such that 퐴 × 푥 = 푏.

퐴 푥 푏

푎1,1 푎1,2 … 푎1,푛 푥1 푏1

푎2,1 푎2,2 … 푎2,푛 푥2 푏2 × = … … … … … …

푎푚,1 푎푚,2 … 푎푚,푛 푥푛 푏푚 Linear systems

Elementary operations. Swap lines

푛 A vector 푥 ∈ ℝ is a solution of a linear system 퐴푚×푛 × 푥 = 푏 iff 푥 is a solution of 퐴′ × 푥 = 푏′, where 퐴′, 푏′ are obtained by swapping, respectively, the lines 푎푖. and 푎푗. in 퐴 and the entries 푏푖 and 푏푗 in 푏 (1 ≤ 푖, 푗 ≤ 푚). Linear systems

Elementary operations. Swap lines

Equivalent system obtained by swapping 푎푖. with 푎푗. and 푏푖 with 푏푗.

푎1,1 푎1,2 … 푎1,푛 푥1 푏1

… … … … 푥2 …

푎 푎 푎 푎푖. → 푖,1 푖,2 … 푖,푛 … 푏푖

… … … … × … = …

푎푗,1 푎푗,2 … 푎푗,푛 … 푏푗 푎푗. →

… … … … … …

푎푚,1 푎푚,2 … 푎푚,푛 푥푛 푏푚 Linear systems

Elementary operations. Scalar multiplication

푛 A vector 푥 ∈ ℝ is a solution of a linear system 퐴푚×푛 × 푥 = 푏 iff 푥 is a solution of 퐴′ × 푥 = 푏′, where 퐴′, 푏′ are obtained by multiplying the line 푎푖. in 퐴 and the entry 푏푖 in 푏 by a scalar 휆 ≠ 0 (1 ≤ 푖 ≤ 푛).

푎1,1 푎1,2 … 푎1,푛 푥1 푏1

… … … … 푥2 …

휆 × 푎 휆 × 푎 휆 × 푎 휆 × 푏 휆 × 푎푖. → 푖,1 푖,2 … 푖,푛 × … = 푖

… … … … … …

푎푚,1 푎푚,2 … 푎푚,푛 푥푛 푏푚 Linear systems

Elementary operations. Line substitution

푛 A vector 푥 ∈ ℝ is a solution of a linear system 퐴푚×푛 × 푥 = 푏 iff 푥 is a solution of 퐴′ × 푥 = 푏′, where 퐴′, 푏′ are obtained by replacing the line 푎푖. in 퐴 and the entry 푏푖 in 푏 respectively by 푎푖. + 휆푎푗. and 푏푖 + 휆푏푗 (1 ≤ 푖, 푗 ≤ 푛).

푎1,1 푎1,2 … 푎1,푛 푥1 푏1

… … … … 푥2 …

푎푖,1 + 푎푖,2 + 푎푖,푛 + … … 푏푖+휆×푏푗 푎푖. + 휆 × 푎푗. → 휆 × 푎푗,1 휆 × 푎푗,2 휆 × 푎푗,푛 × =

… … … … … …

푎푚,1 푎푚,2 … 푎푚,푛 푥푛 푏푚 Linear systems

Elementary operations. Permutations

푚×푚 A square matrix 푃푚×푚 ∈ ℝ is a PERMUTATION MATRIX if is obtained by the elementary operations over the identity matrix 퐼푚×푚.

푛 Given a permutation matrix 푃푚×푚, a vector 푥 ∈ ℝ is a solution ′ ′ of a linear system 퐴푚×푛 × 푥 = 푏 iff 푥 is a solution of 퐴 × 푥 = 푏 , where 퐴′ = 푃 × 퐴 and 푏′ = 푃 × 푏. Linear systems

Solving linear systems. Gauss method

Given a linear system 퐴 × 푥 = 푏, apply elementary operations to obtain an equivalent system 퐴′ × 푥 = 푏′, where 퐴′ is upper triangular and the elements of the diagonal are equal to 1. Then, use back substitution to find the entries of 푥. Linear systems

Solving linear systems. Inverse

Given a linear system 퐴 × 푥 = 푏 if 퐴 is invertible, then 푥 = 퐴−1 × 푏

−1 푡ℎ 퐴 × 푎 .푖 = 퐼.푖, where 퐼.푖 is the 푖 column of the identity. Thus, 푡ℎ −1 the 푖 of 퐴 can be computed by solving 퐴 × 푥 = 퐼.푖. Linear systems

Solving linear systems. Cramer’s rule

Given a linear system 퐴 × 푥 = 푏 if 퐴 is invertible, then 푥 = 푑푒푡(퐵푖) 퐵 퐴 푖 푑푒푡(퐴) , where 푖 is obtained by replacing in the column 푎.푖 with b. Linear systems

Solving linear systems

Given a linear system 퐴푚×푛 × 푥 = 푏, consider the matrix (퐴, 푏)푚×(푛+1), whose first 푚 rows are the rows from 퐴 and the last row is the vector 푏. Then:

▶ If 푟푎푛푘(퐴, 푏) > 푟푎푛푘(퐴), then the linear system has no solution. ▶ If 푟푎푛푘(퐴, 푏) = 푟푎푛푘(퐴) = 푛, then the linear system has exactly one solution. ▶ If 푟푎푛푘(퐴, 푏) = 푟푎푛푘(퐴) < 푛, then the linear system has an infinite number of solutions Exercises Exercises

Exercise 1.

Given 푆 ⊆ ℝ푛 demonstrate that:

푆 is a linear subspace ⇒ 푆 is an affine subspace. Exercises

Exercise 2.

Demonstrate that:

If 푆 ⊆ ℝ푛 is a convex set, not necessarily 푆 is an affine subspace. Exercises

Exercise 3.

Given the set of vector 푆 ⊆ ℝ푛 Demonstrate that:

▶ If the vectors of 푆 are linearly independent, then they are also affine independent.

▶ If the vector of 푆 are affine independent, not necessarily they are linear independent. Exercises

Exercise 4.

Demonstrate that:

If 푆 ⊆ ℝ푛 is a basis of ℝ푛, then for any vector 푣 ∈ ℝ푛 there exists exactly one linear combination of the vectors of 푆 equals to 푣. Exercises

Exercise 5.

Demonstrate the change of basis property. Exercises

Exercise 6.

Suppose are given two matrices 퐴 and 퐵, such that it is possibly to multiply 퐴 × 퐵 and also 퐵 × 퐴. Demonstrate that not necessarily 퐴 × 퐵 = 퐵 × 퐴. Exercises

Exercise 7.

Demonstrate that:

▶ (퐴푡)푡 = 퐴.

▶ (퐴 + 퐵)푡 = 퐴푡 + 퐵푡.

▶ (퐴 × 퐵)푡 = 퐵푡 × 퐴푡. Exercises

Exercise 8.

Given an invertible matrix 퐴 ∈ ℝ푛×푛, demonstrate that:

−1 ▶ 퐴−1 is unique, invertible and (퐴−1) = 퐴.

푡 ▶ 퐴푡 is invertible and (퐴푡)−1 = (퐴−1) .

▶ If 퐵 ∈ ℝ푛×푛 is invertible, then 퐴 × 퐵 is invertible and (퐴 × 퐵)−1 = 퐵−1 × 퐴−1. Exercises

Exercise 9.

Demonstrate that:

A square matrix 퐴 is invertible iff 푑푒푡(퐴) ≠ 0. Linear algebra

Integer linear programming: formulations, techniques and applications.

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Santiago Valdés Ravelo [email protected] August, 2020