MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

1 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS1. SYSTEMS OFEQUATIONS of a linear general system of m EQUATIONS in n unknowns can bewrittena11x1+a12x2+ +a1nxn=b1a21x1+a22x2+ +a2nxn=b2a31x1+a32x2+ +a3nxn= +..+ +..=..am1x1+am2x2+ +amnxn=bm(1)In this system , the aij s and bi s are given real numbers; aijis the coefficient for the unknown xjin the ith equation. We call the set of all aij s arranged in a rectangular array the coefficient matrixof the system . Using MATRIX notation we can write the system asAx=b a11a12 a1na21a22 a2na31a32 amn = (2)We define the augmented coefficient MATRIX for the system as A= a11a12 a1nb1a21a22 a2nb2a31a32 amnbm (3) form of a zeroes in the row of a row of a MATRIX is said to have k leading zeroes if thefirst k elements of the row are all zeroes and the (k+1)

2 Th element of the row is not echelon form of a MATRIX is in row echelon form if each row has more leadingzeroes than the row preceding : August 20, ALGEBRA AND SYSTEMS OF of row echelon following matrices are all in row echelon formA= 347052004 B= 101002000 (4)C= 131041003000 first non-zero element in each row of a MATRIX in row-echelon form is called apivot. For the MATRIX A above the pivots are 3,5,4. For the MATRIX B they are 1,2 and for C they are1,4,3. For the matrices B and C there is no pivot in the last row echelon row echelon MATRIX in which each pivot is a 1 and in which eachcolumn containing a pivot contains no other nonzero entries, is said to be in reduced row echelonform.

3 This implies that columns containing pivots are columns of an identity MATRIX . The matricesD and E below are in reduced row echelon 100010001 (5)E= 100010000 The MATRIX F is in row echelon form but not reduced row echelon 01503000110000100000 (6) number of non-zero rows in the row echelon form of a MATRIX A produced byelementary operations on A is called the rank of A. MATRIX D in equation (5) has rank 3, MATRIX Ehas rank 2, while MATRIX F in (6) has rank to EQUATIONS (stated without proof).

4 A:A system of linear EQUATIONS with coefficient MATRIX A, right hand side vector b, and aug-mented MATRIX Ahas a solution if and only ifrank(A)=rank( A) MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS3b:A linear system of EQUATIONS must have either no solution, one solution, or infinitely :If a linear system has exactly one solution, then the coefficient MATRIX A has at least as manyrows as columns. A system with a unique solution must have at least as many EQUATIONS :If a system of linear EQUATIONS has more unknowns than EQUATIONS , it must either have nosolution or infinitely many :A coefficient MATRIX isnonsingular, that is, the corresponding linear system has one andonly one solution for every choice of right hand side b1,b2.

5 ,bm, if and only ifnumber of rows of A=number of columns of A=rank(A) of linear EQUATIONS and simple 2x2 SYSTEMS using elementary row the following simple 2x2system of linear equationsa11x1+a12x2=b1(7)a21x1+a22x2=b2 We can write this in MATRIX form asAx=bA=[a11a12a21a22],x=[x1x2],b=[b1b2] .(8)If we append the column vector b to the MATRIX A, we obtain the augmented MATRIX for thesystem. This is written as A=[a11a12b1a21a22b2](9)We can perform row operations on this MATRIX to reduce it to reduced row echelon form.

6 We willdo this in steps. The first step is to divide each element of the first row by a11. This will give A1=[1a12a11b1a11a21a22b2](10)Now multiply the first row by a21to yield[a21a21a12a11a21b1a11]and subtract it from the second row4 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONSa21a22b2a21a21a12a11a21b1a11 0a22 a21a12a11b2 a21b1a11 This will give a new MATRIX on which to operate. A2=[1a12a11b1a110a11a22 a21a12a11a11b2 a21b1a11]Now multiply the second row bya11a11a22 a21a12to obtain A3=[1a12a11b1a1101a11b2 a21b1a11a22 a21a12](11)Now multiply the second row bya12a11and subtract it from the first row.

7 First multiply the secondrow bya12a11to yield.[0a12a11a12(a11b2 a21b1)a11(a11a22 a21a12)](12)Now subtract the expression in equation 12 from the first row of A3to obtain the following row.[1a12a11b1a11] [0a12a11a12(a11b2 a21b1)a11(a11a22 a21a12)]=[10b1a11 a12(a11b2 a21b1)a11(a11a22 a21a12)](13)Now replace the first row in A3with the expression in equation 13 of obtain A4 A4=[10b1a11 a12(a11b2 a21b1)a11(a11a22 a21a12)01a11b2 a21b1a11a22 a21a12](14)This can be simplified as by putting the upper right hand term over a common denominator,and canceling like terms as follows A4= 10b1a211a22 b1a11a21a12 a12a211b2+a11a12a21b1a211(a11a22 a21a12)

8 01a11b2 a21b1a11a22 a21a12 = 10b1a211a22 a12a211b2a211(a11a22 a21a12)01a11b2 a21b1a11a22 a21a12 (15)= 10b1a22 a12b2a11a22 a21a1201a11b2 a21b1a11a22 a21a12 We can now read off the solutions for x1and x2. They areMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS5x1=b1a22 a12b2a11a22 a21a12(16)x2=a11b2 a21b1a11a22 a21a12 Each of the fractions has a common denominator. It is called thedeterminantof the MATRIX a 2x2 MATRIX A is given bydet (A)=|A|=a11a22 a21a12(17)If we look closely we can see that we can write the numerator of each expression as a determinantalso.

9 In particularx1=b1a22 a12b2a11a22 a21a12= b1a12b2a22 |A|(18)x2=a11b2 a21b1a11a22 a21a12= a11b1a21b2 |A|The MATRIX of which we compute the determinant in the numerator of the first expression isthe MATRIX A, where the first column has been replaced by the b vector. The MATRIX of which wecompute the determinant in the numerator of the second expression is the MATRIX A where thesecond column has been replaced by the b vector. This procedure for solving SYSTEMS of equationsis calledCramer s ruleand will be discussed in more detail example problem with Cramer s the system of equations3x1+5x2=11(19)8x1 3x2=13 Using Cramer s rulex1= b1a12b2a22 |A|= 11 513 3 358 3 =( 33) (65)( 9) (40)= 98 49=2(20)x2= a11b1a21b2 |A|= 311813 358 3 =(39) (88)( 9) (40)= 49 49=12.

10 And analytical ALGEBRA AND SYSTEMS OF of a determinant of an nxn MATRIX A = aij , written|A|, is definedto be the number computed from the following sum where each element of the sum is the productof n elements:|A|= ( ) ,(21)the sum being taken over all n! permutations of the second subscripts. A term is assigned aplus sign if (i,j,..,r) is an even permutation of (1,2,..,n) and a minus sign if it is an odd even permutation is defined as making an even number of switches of the indices in (1,2.)