# Algebra

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Chapter I MATRIX ALGEBRA AND S Y S T E MS OF L I N EA R EQUATIONS
Goals
To review some known concepts about matrices and determinants. To study elementary operations. To apply elementary operations to calculate the rank and the inverse of a matrix, and to solve systems of linear equations.

I.1.

MATRIX OPERATIONS. REVIEW

We will see some definitions before reviewing matrix operations.I.1.1. Definitions
Definition. A matrix A is a rectangular array of mxn elements (or entries) in a field K, usually R or C. These elements are arranged in m rows and n columns, and are usually denoted by a letter with two subscripts (aij in this case), where i denotes de row and j denotes de column. The set of all mxn matrices in a field K is denoted Emxn (K) or just Emxn.

Definition. Twomatrices A and B are said to be the same type if they have the same size (the same number of rows and columns). Definition. Two matrices A and B of the same type are said to be equal if: aij = bij ∀i = 1, 2,...,m ∧ ∀j = 1, 2, ..., n.

Definition. An mxn matrix whose entries are all zero is a zero (or null) matrix, and it is denoted (0)mxn.

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Matrix algebra and systems of linearequations Definition. The negative of a matrix A = (aij)mxn is the matrix -A = (-aij)mxn. Definition. A matrix with only one row is called a row matrix. A matrix with only one column is called a column matrix. Definition. A matrix is said to be square if it has the same number of rows and columns. Definition. The aii entries of a square matrix A form the main diagonal of this matrix. They are calleddiagonal entries. Definition. A diagonal matrix is a square matrix whose nondiagonal elements are zero. Definition. A scalar matrix is a diagonal matrix, in which all the diagonal entries are equal. Definition. A square matrix with 1’s on the main diagonal and 0’s elsewhere is called an identity (or unit) matrix.

⎡1 0 ... 0 ⎤ ⎢ ⎥ ⎢0 1 ... 0 ⎥ = (δ ) = ⎧1 if I= ⎨ ij ⎢ . . ... . ⎥ ⎩ 0 if ⎢ ⎥ 0 0 ...1 ⎦ ⎣

i= j i≠ j

Definition. A square matrix is said to be upper triangular if all its entries below the main

diagonal are 0’s. Similarly, it is said to be lower triangular if all its entries above the main diagonal are 0’s.

Example
⎛ 1 2 3⎞ ⎜ ⎟ ⎜0 5 6⎟ ⎜0 0 8⎟ ⎝ 24 ⎠ 14 3 ⎛1 0 0 0 ⎞ ⎜ ⎟ ⎜2 3 0 0 ⎟ ⎜4 5 6 0 ⎟ ⎜ ⎟ 9 10 ⎝ 7 8 244 ⎠ 144 3
lower triangular matrix

upper triangularmatrix

Definition. The trace of a square matrix is the sum of the diagonal entries, that is,

tr ( A) = ∑ aii = a11 + a22 + ... + ann .
i =1

n

Let Emxn (K) be the set of all mxn matrices in K.

Definition. Let A and B be two mxn matrices, i.e, A,B∈Emxn. Then the sum A+B is the mxn

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Matrix algebra and systems of linear equations matrix C whoseentries are the sum of the corresponding entries of A and B. That is, if A = (aij) and B = (bij) cij = aij + bij , ∀i = 1,2,...,m ∧ j = 1,2,...,n

Properties
1- Internal operation: ∀A,B∈Emxn ⇒ A+B∈Emxn 2- Associative: ∀A,B,C ∈ Emxn ⇒ (A+B)+C =A+(B + C) 3- Neutral element: ∀A∈Emxn ∃!(0)∈Emxn / A +(0)=(0)+A=A 4- Inverse element: ∀A ∈Emxn ∃(-A)∈Emxn / A+(-A)=(-A)+A=(0) 5- Commutative: ∀A, B∈Emxn ⇒A+B=B+A That is, the set Emxn along with the operation sum is a commutative or abelian group.

I.1.3. Scalar multiples of matrices
Definition. If α is a scalar (α ∈ K) and A is an mxn matrix (A=(aij)∈Emxn), then the scalar

multiple of A is denoted by αA, and is the matrix whose entries are α times the corresponding entries of A. This operation is an external operation.

Properties
1- α⋅(A +B) = α ⋅ A + α ⋅ B
∀ α ∈ K ∧ ∀ A, B ∈ Emxn

2- ( α + β ) ⋅ A = α ⋅ A + β ⋅ A

∀ α, β ∈ K ∧ ∀ A ∈ Emxn

3- ( α⋅β ) ⋅ A = α ⋅ ( β⋅A )

∀ α, β ∈ K ∧ ∀ A ∈ Emxn

4- 1 ⋅ A = A

1∈K

∀ A ∈ Emxn

A nonempty set of elements V on which two operations, addition and scalar multiplication, are defined is said to form a Vector Space if the addition is a commutative group and the scalar...