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**Matrices and Systems of Equations**

Chapter 1 Matrices and Systems of Equations

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**Systems of Linear Equations**

Where the aij’s and bi’s are all real numbers, xi’s are variables . We will refer to systems of the form (1) as m×n linear systems.

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**Definition Example Inconsistent : A linear system has no solution.**

Consistent : A linear system has at least one solution. Example (ⅰ) x1 + x2 = 2 x1 − x2 = 2 (ⅱ) x1 + x2 = 2 x1 + x2 =1 (ⅲ) x1 + x2 = 2 −x1 − x2 =-2

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**Three Operations that can be used on a system **

Definition Two systems of equations involving the same variables are said to be equivalent if they have the same solution set. Three Operations that can be used on a system to obtain an equivalent system: Ⅰ. The order in which any two equations are written may be interchanged. Ⅱ. Both sides of an equation may be multiplied by the same nonzero real number. Ⅲ. A multiple of one equation may be added to (or subtracted from) another.

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**is in strict triangular form.**

n×n Systems Definition A system is said to be in strict triangular form if in the kth equation the coefficients of the first k-1 variables are all zero and the coefficient of xk is nonzero (k=1, …,n). Example The system is in strict triangular form.

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**Example Solve the system**

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**Elementary Row Operations:**

Ⅰ. Interchange two rows. Ⅱ. Multiply a row by a nonzero real number. Ⅲ. Replace a row by its sum with a multiple of another row. Example Solve the system

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2 Row Echelon Form pivotal row pivotal row

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**Definition A matrix is said to be in row echelon form**

ⅰ. If the first nonzero entry in each nonzero row is 1. ⅱ. If row k does not consist entirely of zeros, the number of leading zero entries in row k+1 is greater than the number of leading zero entries in row k. ⅲ. If there are rows whose entries are all zero, they are below the rows having nonzero entries.

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**Example Determine whether the following matrices are**

in row echelon form or not.

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**Definition Definition**

The process of using operations Ⅰ, Ⅱ, Ⅲ to transform a linear system into one whose augmented matrix is in row echelon form is called Gaussian elimination. Definition A linear system is said to be overdetermined if there are more equations than unknows. A system of m linear equations in n unknows is said to be underdetermined if there are fewer equations than unknows (m<n).

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Example

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**Definition A matrix is said to be in reduced row echelon form if:**

ⅰ. The matrix is in row echelon form. ⅱ. The first nonzero entry in each row is the only nonzero entry in its column.

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Homogeneous Systems A system of linear equations is said to be homogeneous if the constants on the right-hand side are all zero. Theorem An m×n homogeneous system of linear equations has a nontrivial solution if n>m.

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3 Matrix Algebra Matrix Notation

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Vectors row vector 1×n matrix n×1 matrix column vector

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**Scalar Multiplication**

Definition Two m×n matrices A and B are said to be equal if aij=bij for each i and j. Scalar Multiplication If A is a matrix and k is a scalar, then kA is the matrix formed by multiplying each of the entries of A by k. Definition If A is an m×n matrix and k is a scalar, then kA is the m×n matrix whose (i, j) entry is kaij.

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**Matrix Addition Definition**

Two matrices with the same dimensions can be added by adding their corresponding entries. Definition If A=(aij) and B=(bij) are both m×n matrices,then the sum A+B is the m×n matrix whose (i, j) entry is aij+bij for each ordered pair (i, j).

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Example Let Then calculate 。

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**cij = ai1b1j + ai2b2j +…+ ainbnj = aikbkj.**

Matrix Multiplication Definition If A=(aij) is an m×n matrix and B=(bij) is an n×r matrix, then the product AB=C=(cij) is the m×r matrix whose entries are defined by cij = ai1b1j + ai2b2j +…+ ainbnj = aikbkj. k=1 n

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Example then calculate AB. 1. If then calculate AB and BA. 2. If

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**Matrix Multiplication and Linear Systems**

Case 1 One equation in Several Unknows If we let and then we define the product AX by

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**Case 2 M equations in N Unknows**

If we let and then we define the product AX by

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Definition If a1, a2, … , an are vectors in Rm and c1, c2, … , cn are scalars, then a sum of the form c1a1+c2a2+‥‥cnan is said to be a linear combination of the vectors a1, a2, … , an . Theorem (Consistency Theorem for Linear Systems) A linear system AX=b is consistent if and only if b can be written as a linear combination of the column vectors of A.

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Theorem Each of the following statements is valid for any scalars k and l and for any matrices A, B and C for which the indicated operations are defined. A+B=B+A (A+B)+C=A+(B+C) (AB)C=A(BC) A(B+C)=AB+AC (A+B)+C=AC+BC (kl)A=k(lA) k(AB)=(kA)B=A(kB) (k+l)A=kA+lA k(A+B)=kA+kB

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The Identity Matrix Definition The n×n identity is the matrix where

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**Matrix Inversion Definition Definition**

An n×n matrix A is said to be nonsingular or invertible if there exists a matrix B such that AB=BA=I. Then matrix B is said to be a multiplicative inverse of A. Definition An n×n matrix is said to be singular if it does not have a multiplicative inverse.

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**The Transpose of a Matrix**

Theorem If A and B are nonsingular n×n matrices, then AB is also nonsingular and (AB)-1=B-1A-1 The Transpose of a Matrix Definition The transpose of an m×n matrix A is the n×m matrix B defined by bji=aij for j=1, …, n and i=1, …, m. The transpose of A is denoted by AT.

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**Algebra Rules for Transpose:**

(AT)T=A (kA)T=kAT (A+B)T=AT+BT (AB)T=BTAT Definition An n×n matrix A is said to be symmetric if AT=A.

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4. Elementary Matrices If we start with the identity matrix I and then perform exactly one elementary row operation, the resulting matrix is called an elementary matrix.

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**Type I. An elementary matrix of type I is a matrix obtained by**

interchanging two rows of I. Example Let and let A be a 3×3 matrix then

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**Type II. An elementary matrix of type II is a matrix obtained by**

multiplying a row of I by a nonzero constant. Example Let and let A be a 3×3 matrix then

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**Type III. An elementary matrix of type III is a matrix obtained**

from I by adding a multiple of one row to another row. Example Let and let A be a 3×3 matrix

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**In general, suppose that E is an n×n elementary matrix**

In general, suppose that E is an n×n elementary matrix. E is obtained by either a row operation or a column operation. If A is an n×r matrix, premultiplying A by E has the effect of performing that same row operation on A. If B is an m×n matrix, postmultiplying B by E is equivalent to performing that same column operation on B.

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Example Let , Find the elementary matrices ， ，such that

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Theorem If E is an elementary matrix, then E is nonsingular and E-1 is an elementary matrix of the same type. Definition A matrix B is row equivalent to A if there exists a finite sequence E1, E2, … , Ek of elementary matrices such that B=EkEk-1‥‥E1A

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**Theorem 1.4.2 (Equivalent Conditions for Nonsingularity)**

Let A be an n×n matrix. The following are equivalent: A is nonsingular. Ax=0 has only the trivial solution 0. A is row equivalent to I. Theorem The system of n linear equations in n unknowns Ax=b has a unique solution if and only if A is nonsingular.

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**A method for finding the inverse of a matrix**

If A is nonsingular, then A is row equivalent to I and hence there exist elementary matrices E1, …, Ek such that EkEk-1‥‥E1A=I multiplying both sides of this equation on the right by A-1 EkEk-1‥‥E1I=A-1 row operations Thus (A I) (I A-1)

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Example Compute A-1 if

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**Example Solve the system**

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**Diagonal and Triangular Matrices**

An n×n matrix A is said to be upper triangular if aij=0 for i>j and lower triangular if aij=0 for i<j. A is said to be triangular if it is either upper triangular or lower triangular. An n×n matrix A is said to be diagonal if aij=0 whenever i≠j .

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**5. Partitioned Matrices -2 4 1 3 1 1 1 1 3 2 -1 2 = C= 4 6 2 2 4**

C11 C12 = C21 C22 C= B= =(b1, b2, b3) AB=A(b1, b2, b3)=(Ab1, Ab2, Ab3)

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**In general, if A is an m×n matrix and B is an n×r that has**

been partitioned into columns (b1, …, br), then the block multiplication of A times B is given by AB=(Ab1, Ab2, … , Abr) If we partition A into rows, then Then the product AB can be partitioned into rows as follows:

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**Case 1 B=(B1 B2), where B1 is an n×t matrix and B2**

Block Multiplication Let A be an m×n matrix and B an n×r matrix. Case 1 B=(B1 B2), where B1 is an n×t matrix and B2 is an n×(r-t) matrix. AB= A(b1, … , bt, bt+1, … , br) = (Ab1, … , Abt, Abt+1, … , Abr) = (A(b1, … , bt), A(bt+1, … , br)) = (AB1 AB2)

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**Case 2 A= ,where A1 is a k×n matrix and A2**

is an (m-k)×n matrix. Thus Case 3 A=(A1 A2) and B= , where A1 is an m×s matrix and A2 is an m×(n-s) matrix, B1 is an s×r matrix and B2 is an (n-s)×r matrix. Thus

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**Case 4 Let A and B both be partitioned as follows：**

B11 B s B= B21 B n-s t r-t A11 A k A= A21 A m-k s n-s Then

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In general, if the blocks have the proper dimensions, the block multiplication can be carried out in the same manner as ordinary matrix multiplication.

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Example Let Then calculate AB.

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Example Let A be an n×n matrix of the form where A11 is a k×k matrix (k<n). Show that A is nonsingular if and only if A11 and A22 are nonsingular.

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