Matrices and Systems of Equations Chapter 1 Matrices and Systems of Equations

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.

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

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.

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.

Example Solve the system

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

2 Row Echelon Form pivotal row pivotal row

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.

Example Determine whether the following matrices are in row echelon form or not.

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).

Example

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.

Homogeneous Systems A system of linear equations is said to be homogeneous if the constants on the right-hand side are all zero. Theorem 1.2.1 An m×n homogeneous system of linear equations has a nontrivial solution if n>m.

3 Matrix Algebra Matrix Notation

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

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.

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).

Example Let Then calculate 。

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

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

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

Case 2 M equations in N Unknows If we let and then we define the product AX by

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 1.3.1 (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.

Theorem 1.3.2 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

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

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.

The Transpose of a Matrix Theorem 1.3.3 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.

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.

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.

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

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

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

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.

Example Let , Find the elementary matrices ， ，such that .

Theorem 1.4.1 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

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 1.4.3 The system of n linear equations in n unknowns Ax=b has a unique solution if and only if A is nonsingular.

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)

Example Compute A-1 if

Example Solve the system

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 .

5. Partitioned Matrices -2 4 1 3 1 1 1 1 3 2 -1 2 = C= 4 6 2 2 4 -2 4 1 3 1 1 1 1 3 2 -1 2 4 6 2 2 4 C11 C12 = C21 C22 C= -1 2 1 B= 2 3 1 1 4 1 =(b1, b2, b3) AB=A(b1, b2, b3)=(Ab1, Ab2, Ab3)

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:

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)

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

Case 4 Let A and B both be partitioned as follows： B11 B12 s B= B21 B22 n-s t r-t A11 A12 k A= A21 A22 m-k s n-s Then

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

Example Let Then calculate AB.

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.