ࡱ>  |Q'` qbjbj{P{P ;d::Rh.fffz"a"a"a8ZacDz,.e|(F|F|F|b~:ւP$<hfv^~b~vvF|F|v*F|fF|v.?,:f_F|"e w"ad @0Kx^x_xf_<&|b&&&"^&&&vvvvzzzdF]Dzzz]zzz  Chapter 04.01 Introduction to Matrix Algebra After reading this chapter, you should be able to  HYPERLINK \l "_What_is_a_matrix?" know what a matrix XE "matrix"  is,  HYPERLINK \l "_What_are_the_special types of matri" identify special types of matrices, XE "special types of matrices"   HYPERLINK \l "_When_are_two_matrices considered to" know when two matrices XE "matrices"  are equal, add, subtract and multiply matrices, rules of binary matrix operations, find transpose of a matrix, find inverse of a matrix and its application to solving simultaneous linear equations. What is a matrix XE "matrix" ? Matrices XE "Matrices"  are everywhere. If you have used a spreadsheet such as Excel or written a table, you have used a matrix XE "matrix" . Matrices make presentation of numbers clearer and make calculations easier to program. Look at the matrix below about the sale of tires in a Blowoutrus store given by quarter and make of tires. Quarter 1Quarter 2Quarter 3Quarter 4 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 If one wants to know how many Copper tires were sold in Quarter 4, we go along the row Copper and column Quarter 4 and find that it is 27. So what is a matrix XE "matrix" ? A matrix XE "matrix"  is a rectangular array of elements. The elements can be symbolic expressions or numbers. Matrix XE "Matrix"  [A] is denoted by  EMBED Equation.3  Row  EMBED Equation.3  of [A] has  EMBED Equation.3  elements and is  EMBED Equation.3  and Column  EMBED Equation.3 of [A] has  EMBED Equation.3  elements and is  EMBED Equation.3  Each matrix XE "matrix"  has rows and columns and this defines the size of the matrix. If a matrix [A] has  EMBED Equation.3 rows and  EMBED Equation.3 columns, the size of the matrix is denoted by EMBED Equation.3 . The matrix [A] may also be denoted by  EMBED Equation.3  to show that  EMBED Equation.3  is a matrix with  EMBED Equation.3  rows and  EMBED Equation.3  columns. Each entry XE "entry"  in the matrix XE "matrix"  is called the entry or element of the matrix XE "element of the matrix"  and is denoted by  EMBED Equation.3  where  EMBED Equation.3  is the row number and  EMBED Equation.3  is the column number of the element. The matrix XE "matrix"  for the tire sales example could be denoted by the matrix [A] as  EMBED Equation.3  There are 3 rows and 4 columns, so the size of the matrix XE "matrix"  is  EMBED Equation.3 . In the above  EMBED Equation.3  matrix,  EMBED Equation.3 . What are the special types of matrices XE "special types of matrices" ? Vector XE "Vector" : A vector XE "vector"  is a matrix XE "matrix"  that has only one row or one column. There are two types of vectors row vector XE "row vector" s and column vector XE "column vector" s. Row vector XE "Row vector"  XE "vector" : If a matrix XE "matrix"  has one row, it is called a row vector XE "row vector"   EMBED Equation.3  and  EMBED Equation.3  is the dimension of the row vector XE "row vector" . Example 1 Give an example of a row vector XE "row vector" . Solution [B] = [25 20 3 2 0] is an example of a row vector XE "row vector"  of dimension 5. Column vector XE "Column vector"  XE "vector" : If a matrix XE "matrix"  has one column, it is called a column vector XE "column vector"   EMBED Equation.3  and  EMBED Equation.3  is the dimension of the vector XE "vector" . Example 2 Give an example of a column vector XE "column vector" . Solution  EMBED Equation.3  is an example of a column vector XE "column vector"  of dimension 3. Square matrix XE "Square matrix"  XE "matrix" : If the number of rows ( EMBED Equation.3 ) of a matrix is equal to the number of columns ( EMBED Equation.3 ) of the matrix, ( EMBED Equation.3 ), it is called a square matrix XE "square matrix" . The entries  EMBED Equation.3  are called the diagonal XE "diagonal"  elements XE "diagonal elements"  of a square matrix. Sometimes the diagonal of the matrix is also called the principal XE "principal"  or main of the matrix. Example 3 Give an example of a square matrix XE "square matrix" . Solution  EMBED Equation.3  is a square matrix XE "square matrix"  as it has same number of rows and columns, that is, three. The diagonal XE "diagonal"  elements XE "diagonal elements"  of [A] are EMBED Equation.3 . Upper triangular matrix XE "Upper triangular matrix"  XE "matrix" : A  EMBED Equation.3  matrix for which  EMBED Equation.3  is called an upper triangular matrix XE "upper triangular matrix" . That is, all the elements below the diagonal XE "diagonal"  entries are zero. Example 4 Give an example of an upper triangular matrix XE "upper triangular matrix" . Solution  EMBED Equation.3  is an upper triangular matrix XE "upper triangular matrix" . Lower triangular matrix XE "Lower triangular matrix"  XE "matrix" : A  EMBED Equation.3  matrix for which  EMBED Equation.3  is called a lower triangular matrix XE "lower triangular matrix" . That is, all the elements above the diagonal XE "diagonal"  entries are zero. Example 5 Give an example of a lower triangular matrix XE "lower triangular matrix" . Solution  EMBED Equation.3  is a lower triangular matrix XE "lower triangular matrix" . Diagonal XE "Diagonal"  matrix XE "Diagonal matrix"  XE "matrix" : A square matrix XE "square matrix"  with all non-diagonal XE "diagonal"  elements XE "diagonal elements"  equal to zero is called a diagonal matrix XE "diagonal matrix" , that is, only the diagonal entries of the square matrix can be non-zero, ( EMBED Equation.3 ). Example 6 Give examples of a diagonal XE "diagonal"  matrix XE "diagonal matrix" . Solution  EMBED Equation.3  EMBED Equation.3  is a diagonal XE "diagonal"  matrix XE "diagonal matrix" . Any or all the diagonal XE "diagonal"  entries of a diagonal matrix XE "diagonal matrix"  can be zero. For example  EMBED Equation.3  is also a diagonal XE "diagonal"  matrix XE "diagonal matrix" . Identity matrix XE "Identity matrix"  XE "matrix" : A diagonal XE "diagonal"  matrix XE "diagonal matrix"  with all diagonal elements XE "diagonal elements"  equal to one is called an identity matrix XE "identity matrix" , ( EMBED Equation.3 ; and  EMBED Equation.3  for all  EMBED Equation.3 ). Example 7 Give an example of an identity matrix XE "identity matrix" . Solution [A] = EMBED Equation.3  is an identity matrix XE "identity matrix" . Zero matrix XE "Zero matrix"  XE "matrix" : A matrix whose all entries are zero is called a zero matrix XE "zero matrix" , ( EMBED Equation.3  for all  EMBED Equation.3  and  EMBED Equation.3 ). Example 8 Give examples of a zero matrix XE "zero matrix" . Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  are all examples of a zero matrix XE "zero matrix" . Tridiagonal matrices XE "Tridiagonal matrices"  XE "matrices" : A tridiagonal matrix XE "matrix"  is a square matrix XE "square matrix"  in which all elements not on the major diagonal XE "diagonal" , the diagonal above the major diagonal and the diagonal below the major diagonal are zero. Example 9 Give an example of a tridiagonal matrix XE "matrix" . Solution  EMBED Equation.3  is a tridiagonal matrix XE "matrix" . When are two matrices XE "matrices"  considered to be equal? Two matrices XE "matrices"  [A] and [B] are equal if the size of [A] and [B] is the same (number of rows and columns are same for [A] and [B]) and  EMBED Equation.3  for all  EMBED Equation.3  and  EMBED Equation.3 . Example 10 What would make  EMBED Equation.3 to be equal to  EMBED Equation.3 , Solution The two matrices XE "matrices"   EMBED Equation.3  and  EMBED Equation.3  would be equal if  EMBED Equation.3  How do you add two matrices XE "matrices" ? Two matrices XE "matrices"  [A] and [B] can be added only if they are the same size, then the addition is shown as [C] = [A] + [B] where  EMBED Equation.3  Example 11 Add two matrices XE "matrices"   EMBED Equation.3   EMBED Equation.3  Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Example 12 Blowout rus store has two locations A and B, and their sales of tires are given by make (in rows) and quarters (in columns) as shown below.  EMBED Equation.3   EMBED Equation.3  where the rows represent sale of Tirestone, Michigan and Copper tires and the columns represent the quarter number - 1, 2, 3, 4. What are the total sales of the two locations by make and quarter? Solution  EMBED Equation.3  = EMBED Equation.3 + EMBED Equation.3  = EMBED Equation.3   EMBED Equation.3  So if one wants to know the total number of Copper tires sold in quarter 4 in the two locations, we would look at Row 3 Column 4 to give  EMBED Equation.3  How do you subtract two matrices XE "matrices" ? Two matrices XE "matrices"  [A] and [B] can be subtracted only if they are the same size and the subtraction is given by [D] = [A] [B] where  EMBED Equation.3  Example 13 Subtract matrix XE "matrix"  [B] from matrix [A].  EMBED Equation.3   EMBED Equation.3  Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Example 14 Blowout rus store has two locations A and B and their sales of tires are given by make (in rows) and quarters (in columns) as shown below.  EMBED Equation.3   EMBED Equation.3  where the rows represent sale of Tirestone, Michigan and Copper tires and the columns represent the quarter number- 1, 2, 3, 4. How many more tires did store A sell than store B of each brand in each quarter? Solution  EMBED Equation.3  = EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  So if you want to know how many more Copper Tires were sold in quarter 4 in Store A than Store B,  EMBED Equation.3  Note that  EMBED Equation.3  implying that store A sold 1 less Michigan tire than Store B in quarter 3. How do I multiply two matrices XE "matrices" ? Two matrices XE "matrices"  [A] and [B] can be multiplied only if the number of columns of [A] is equal to the number of rows of [B] to give  EMBED Equation.3  If [A] is a  EMBED Equation.3  matrix XE "matrix"  and [B] is a  EMBED Equation.3  matrix, the resulting matrix [C] is a  EMBED Equation.3  matrix. So how does one calculate the elements of [C] matrix XE "matrix" ?  EMBED Equation.3   EMBED Equation.3  for each  EMBED Equation.3 . To put it in simpler terms, the  EMBED Equation.3  row and  EMBED Equation.3  column of the [C] matrix XE "matrix"  in [C] = [A][B] is calculated by multiplying the  EMBED Equation.3  row of [A] by the  EMBED Equation.3  column of [B], that is,  EMBED Equation.3   EMBED Equation.3  Example 15 Given  EMBED Equation.3   EMBED Equation.3  find  EMBED Equation.3  Solution  EMBED Equation.3  can be found by multiplying the first row of [A] by the second column of [B],  EMBED Equation.3   EMBED Equation.3  Similarly, one can find the other elements of [C] to give  EMBED Equation.3  Example 16 Blowout rus store location A and the sales of tires are given by make (in rows) and quarters (in columns) as shown below  EMBED Equation.3  where the rows represent sale of Tirestone, Michigan and Copper tires and the columns represent the quarter number - 1, 2, 3, 4. Find the per quarter sales of store A if following are the prices of each tire. Tirestone = $33.25 Michigan = $40.19 Copper = $25.03 Solution The answer is given by multiplying the price matrix XE "matrix"  by the quantity sales of store A. The price matrix is  EMBED Equation.3 , then the per quarter sales of store A would be given by  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  = EMBED Equation.3  = $1182.38 Similarly  EMBED Equation.3  So each quarter sales of store A in dollars are given by the four columns of the row vector XE "row vector"   EMBED Equation.3  Remember since we are multiplying a  EMBED Equation.3  matrix XE "matrix"  by a EMBED Equation.3  matrix, the resulting matrix is a  EMBED Equation.3  matrix. What is a scalar product of a constant and a matrix XE "matrix" ? If  EMBED Equation.3  is a  EMBED Equation.3  matrix XE "matrix"  and EMBED Equation.3  is a real number, then the scalar product of  EMBED Equation.3  and [A] is another matrix [B], where  EMBED Equation.3 . Example 17 Let  EMBED Equation.3 . Find 2 [A] Solution  EMBED Equation.3  Then  EMBED Equation.3 =2 EMBED Equation.3  = EMBED Equation.3   EMBED Equation.3  What is a linear combination of matrices XE "linear combination of matrices" ? If  EMBED Equation.3  are matrices XE "matrices"  of the same size and EMBED Equation.3  are scalars, then  EMBED Equation.3  is called a linear combination of  EMBED Equation.3  Example 18 If  EMBED Equation.3  then find  EMBED Equation.3  Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  What are some of the rules of binary matrix XE "matrix"  operations? Commutative law of addition XE "Commutative law of addition"  If [A] and [B] are  EMBED Equation.3  matrices XE "matrices" , then  EMBED Equation.3  Associate law of addition XE "Associate law of addition"  If [A], [B] and [C] all are  EMBED Equation.3  matrices XE "matrices" , then  EMBED Equation.3  Associate law of multiplication XE "Associate law of multiplication"  If [A], [B] and [C] are  EMBED Equation.3  size matrices XE "matrices" , respectively, then  EMBED Equation.3  and the resulting matrix XE "matrix"  size on both sides is  EMBED Equation.3  Distributive law XE "Distributive law"  If [A] and [B] are  EMBED Equation.3  size matrices XE "matrices" , and [C] and [D] are  EMBED Equation.3  size matrices  EMBED Equation.3   EMBED Equation.3  and the resulting matrix XE "matrix"  size on both sides is  EMBED Equation.3  Example 19 Illustrate the associative law of multiplication XE "associative law of multiplication"  of matrices XE "matrices"  using  EMBED Equation.3  Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  The above illustrates the associate law of multiplication of matrices XE "multiplication of matrices" . Is [A][B]=[B][A]? First both operations  EMBED Equation.3  EMBED Equation.3  and  EMBED Equation.3  EMBED Equation.3  are only possible if  EMBED Equation.3  and  EMBED Equation.3  are square matrices XE "matrices"  of same size. Why? If  EMBED Equation.3  EMBED Equation.3  exists, number of columns of  EMBED Equation.3  has to be same as the number of rows of  EMBED Equation.3  and if  EMBED Equation.3  EMBED Equation.3  exists, number of columns of  EMBED Equation.3  has to be same as the number of rows of  EMBED Equation.3 . Even then in general  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 . Example 20 Illustrate if  EMBED Equation.3  EMBED Equation.3 = EMBED Equation.3  EMBED Equation.3  for the following matrices EMBED Equation.3  XE "matrices"   EMBED Equation.3  Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  [A][B] ( [B][A] Transpose XE "Transpose"  of a matrix XE "matrix" : Let [A] be a  EMBED Equation.3  matrix. Then [B] is the transpose XE "transpose"  of the [A] if  EMBED Equation.3  for all  EMBED Equation.3  and  EMBED Equation.3 . That is, the  EMBED Equation.3  row and the  EMBED Equation.3  column element of [A] is the  EMBED Equation.3  row and  EMBED Equation.3  column element of [B]. Note, [B] would be a  EMBED Equation.3  matrix. The transpose of [A] is denoted by  EMBED Equation.3 . Example 21 Find the transpose XE "transpose"  of [A]= EMBED Equation.3  Solution The transpose XE "transpose"  of [A] is  EMBED Equation.3  Note, the transpose XE "transpose"  of a row vector XE "row vector"  is a column vector XE "column vector"  and the transpose of a column vector XE "vector"  is a row vector. Also, note that the transpose XE "transpose"  of a transpose of a matrix XE "matrix"  is the matrix itself, that is,  EMBED Equation.3 . Also,  EMBED Equation.3  . Symmetric matrix XE "Symmetric matrix"  XE "matrix" : A square matrix XE "square matrix"  [A] with real elements where  EMBED Equation.3  for  EMBED Equation.3  and  EMBED Equation.3  is called a symmetric matrix XE "symmetric matrix" . This is same as, if  EMBED Equation.3 , then  EMBED Equation.3  is a symmetric matrix. Example 22 Give an example of a symmetric matrix XE "symmetric matrix" . Solution  EMBED Equation.3  is a symmetric matrix XE "symmetric matrix"  as  EMBED Equation.3  and  EMBED Equation.3 . Matrix XE "Matrix"  algebra is used for solving system of equations XE "system of equations" . Can you illustrate this concept? Matrix XE "Matrix"  algebra is used to solve a system of simultaneous linear equations. In fact, for many mathematical procedures such as solution of set of nonlinear equations, interpolation, integration, and differential equations, the solutions reduce to a set of simultaneous linear equations. Let us illustrate with an example for interpolation. Example 23 The upward velocity of a rocket is given at three different times on the following table Time,  EMBED Equation.3 Velocity,  EMBED Equation.3 sm/s5106.88177.212279.2 The velocity data is approximated by a polynomial as  EMBED Equation.3  Set up the equations in matrix XE "matrix"  form to find the coefficients  EMBED Equation.3  of the velocity profile. Solution The polynomial is going through three data points  EMBED Equation.3  where from the above table  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Requiring that  EMBED Equation.3  passes through the three data points gives  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Substituting the data  EMBED Equation.3  gives  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  or  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  This set of equations can be rewritten in the matrix XE "matrix"  form as  EMBED Equation.3  The above equation can be written as a linear combination as follows  EMBED Equation.3  and further using matrix XE "matrix"  multiplications gives  EMBED Equation.3  The above is an illustration of why matrix XE "matrix"  algebra is needed. The complete solution to the set of equations is given later in this chapter. For a general set of m linear equations and n unknowns,  EMBED Equation.3   EMBED Equation.3  .  EMBED Equation.3  can be rewritten in the matrix XE "matrix"  form as  EMBED Equation.3  Denoting the matrices XE "matrices"  by  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 , the system of equation is  EMBED Equation.3   EMBED Equation.3 = EMBED Equation.3 , where  EMBED Equation.3  is called the coefficient matrix XE "matrix" ,  EMBED Equation.3  is called the right hand side vector XE "vector"  and  EMBED Equation.3  is called the solution vector. EMBED Equation.3  Sometimes  EMBED Equation.3   EMBED Equation.3 = EMBED Equation.3  systems of equations is written in the augmented form. That is  EMBED Equation.3  Can you divide two matrices XE "matrices" ? If [A] [B] = [C] is defined, it might seem intuitive that [A]= EMBED Equation.3 , but matrix XE "matrix"  division is not defined. However an inverse of a matrix XE "inverse of a matrix"  can be defined for certain types of square matrices XE "matrices" . The inverse of a square matrix XE "square matrix"  [A], if existing, is denoted by [A]  EMBED Equation.3  such that [A][A] EMBED Equation.3 = [I] = [A] EMBED Equation.3 [A]. In other words, let [A] be a square matrix XE "square matrix" . If [B] is another square matrix XE "matrix"  of same size such that [B][A] = [I], then [B] is the inverse of [A]. [A] is then called to be invertible XE "invertible"  or nonsingular. If [A]  EMBED Equation.3  does not exist, [A] is called to be noninvertible or singular XE "singular" . If [A] and [B] are two  EMBED Equation.3  matrices XE "matrices"  such that [B] [A] = [I], then these statements are also true [B] is the inverse of [A] [A] is the inverse of [B] [A] and [B] are both invertible XE "invertible"  [A] [B]=[I]. [A] and [B] are both nonsingular all columns of [A] or [B]are linearly independent XE "linearly independent"  all rows of [A] or [B] are linearly independent XE "linearly independent" . Example 24 Show if  EMBED Equation.3  Solution  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  = [I] Since [B] [A] = [I], [B] is the inverse of [A] and [A] is the inverse of [B]. But we can also show that  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  to show that [A] is the inverse of [B]. Can I use the concept of the inverse of a matrix XE "inverse of a matrix"  to find the solution of a set of equations [A] [X] = [C]? Yes, if the number of equations is same as the number of unknowns, the coefficient matrix XE "matrix"  [A] is a square matrix XE "square matrix" . Given [A][X] = [C] Then, if [A] EMBED Equation.3 exists, multiplying both sides by [A] EMBED Equation.3 . [A] EMBED Equation.3  [A][X] = [A] EMBED Equation.3  [C] [I][X] = [A] EMBED Equation.3  [C] [X] = [A] EMBED Equation.3  [C] This implies that if we are able to find [A] EMBED Equation.3 , the solution vector XE "vector"  of [A][X] = [C] is simply a multiplication of [A] EMBED Equation.3  and the right hand side vector, [C]. How do I find the inverse of a matrix XE "inverse of a matrix" ? If [A] is a  EMBED Equation.3  matrix XE "matrix" , then [A] EMBED Equation.3  is a  EMBED Equation.3  matrix and according to the definition of inverse of a matrix XE "inverse of a matrix"  [A][A] EMBED Equation.3  = [I]. Denoting  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Using the definition of matrix XE "matrix"  multiplication, the first column of the [A] EMBED Equation.3  matrix can then be found by solving  EMBED Equation.3  Similarly, one can find the other columns of the [A] EMBED Equation.3  matrix XE "matrix"  by changing the right hand side accordingly. _________________________________ Example 25 The upward velocity of the rocket is given by Time,  EMBED Equation.3 Velocity,  EMBED Equation.3 sm/s5106.88177.212279.2 In an earlier example, we wanted to approximate the velocity profile by  EMBED Equation.3  We found that the coefficients  EMBED Equation.3  are given by  EMBED Equation.3  First find the inverse of  EMBED Equation.3  and then use the definition of inverse to find the coefficients  EMBED Equation.3  Solution If  EMBED Equation.3  is the inverse of [A], Then  EMBED Equation.3  gives three sets of equations  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Solving the above three sets of equations separately gives  EMBED Equation.3 = EMBED Equation.3   EMBED Equation.3 = EMBED Equation.3   EMBED Equation.3  Hence  EMBED Equation.3  Now  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3  Using the definition of  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  So  EMBED Equation.3  If the inverse of a square matrix XE "square matrix"  [A] exists, is it unique? Yes, the inverse of a square matrix XE "square matrix"  is unique, if it exists. The proof is as follows. Assume that the inverse of [A] is [B] and if this inverse is not unique, then let another inverse of [A] exist called [C]. [B] is inverse of [A], then [B][A] = [I] Multiply both sides by [C], [B][A][C] = [I][C] [B][A][C] = [C] Since [C] is inverse of [A], [A][C] = [I] [B][I] = [C] [B] = [C] This shows that [B] and [C] are the same. So inverse of [A] is unique. INTRODUCTION TO MATRIX ALGEBRATopicIntroduction to Matrix AlgebraSummaryKnow what a matrix is; Identify special types of matrices; When two matrices are equal; Add, subtract and multiply matrices; Learn rules of binary operations on matrices; Know what unary operations mean; Find the transpose of a square matrix and it relationship to symmetric matrices; Setup simultaneous linear equations in matrix form and vice-versa; Understand the concept of inverse of a matrix.MajorGeneral EngineeringAuthorsAutar KawDate DATE \@ "MMMM d, yyyy" \* MERGEFORMAT March 23, 2010Web Site HYPERLINK "http://numericalmethods.eng.usf.edu" http://numericalmethods.eng.usf.edu     04.01. PAGE 6 Chapter 04.01 Introduction to Matrix Algebra 04.01. 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