A.2 Cramer's Method The value of each unknown variable in the set of equations is expressed as the ratio of two determinants. If we let N, with an appropriate subscript, represent the numerator determinant and A represent the denominator determinant, then the /cth unknown x k is N, Xk = (A.4) The denominator determinant A is the same for every unknown variable and is called the characteristic determinant of the set of equations. The numerator determinant N k varies with each unknown. Equation A.4 is referred to as Cramer's method for solving simultaneous equations. A.3 The Characteristic Determinant Once we have organized the set of simultaneous equations into an ordered array, as illustrated by Eqs. A.l and A.3, it is a simple matter to form the characteristic determinant. This determinant is the square array made up from the coefficients of the unknown variables. For example, the characteristic determinants of Eqs. A.l and A.3 are A = 21 -3 -8 -9 6 -4 -12 -2 22 (A.5) and A = 2 0 7 -1 4 0 0 3 2 (A.6) respectively. A.4 The Numerator Determinant The numerator determinant N k is formed from the characteristic determi- nant by replacing the kth column in the characteristic determinant with the column of values appearing on the right-hand side of the equations. For example, the numerator determinants for evaluating /j, / 2 , and 1*3 in Eqs. A.l are A/, = -33 50 -9 6 -4 -12 -2 22 (A.7) N-> = 21 -33 -12 -3 3 -2 -8 50 22 (A.8) and N 3 = 21 -3 -8 -9 6 -4 -33 3 50 (A.9) The numerator determinants for the evaluation of t?j, v 2 , and ?; 3 in Eqs. A.3 are N, = 4 16 5 -1 4 0 0 3 2 (A.10) and A r , 2 0 7 4 16 5 0 3 2 7V 3 2 0 7 - 1 4 0 4 16 5 (A.ll) (A.12) A.5 The Evaluation of a Determinant The value of a determinant is found by expanding it in terms of its minors. The minor of any element in a determinant is the determinant that remains after the row and column occupied by the element have been deleted. For example, the minor of the element 6 in Eq. A.7 is •33 -12 50 22 while the minor of the element 22 in Eq. A.7 is -33 3 The cofactor of an element is its minor multiplied by the sign- controlling factor -l (/+/) , where i and j denote the row and column, respectively, occupied by the element. Thus the cofactor of the element 6 in Eq. A.7 is _ j (2+2) -33 -12 50 22 and the cofactor of the clement 22 is _l(3+3) -33 -9 3 6 The cofactor of an element is also referred to as its signed minor. The sign-controlling factor — l ((+y) will equal +1 or —1 depending on whether i + j is an even or odd integer. Thus the algebraic sign of a cofac- tor alternates between + 1 and —1 as we move along a row or column. For a 3 X 3 determinant, the plus and minus signs form the checkerboard pat- tern illustrated here: + + + + - + A determinant can be expanded along any row or column. Thus the first step in making an expansion is to select a row i or a column j. Once a row or column has been selected, each element in that row or column is multi- plied by its signed minor, or cofactor. The value of the determinant is the sum of these products. As an example, let us evaluate the determinant in Eq. A.5 by expanding it along its first column. Following the rules just explained, we write the expansion as A = 21(1) -2 22 - 3(-1) -9 -12 •4 22 -8(1) •9 -12 6 -2 (A.13) The 2X2 determinants in Eq. A.13 can also be expanded by minors. The minor of an element in a 2 x 2 determinant is a single element. It fol- lows that the expansion reduces to multiplying the upper-left element by the lower-right element and then subtracting from this product the product of the lower-left element times the upper-right element. Using this obser- vation, we evaluate Eq. A.13 to A = 21(132 -8)+ 3(-198 - 48) - 8(18 + 72) = 2604 - 738 - 720 = 1146. (A.14) Had we elected to expand the determinant along the second row of ele- ments, we would have written A = -3(-1) -12 22 +6(+1) 21 -8 -12 22 -2(-1) 21 -8 -9 -4 = 3(-198 - 48) + 6(462 - 96) + 2(-84 - 72) -738 + 2196 - 312 = 1146. (A.15) The numerical values of the determinants N u N 2 , and N 3 given by Eqs. A.7, A.8, and A.9 are Ni = 1146, (A.16) and N 2 = 2292, N 3 = 3438. (A.17) (A.18) It follows from Eqs. A.15 through A.18 that the solutions for i\, i 2 , and i 3 in Eq. A.l are H- T -1A, i 2 = -/ = 2A, (A.19) and N 3 h = ~T = 3 A. We leave you to verify that the solutions for v h v 2 , and v 3 in Eqs. A.3 are 49 Vl = —= -9.8 V, v 2 = ^| = -23.6 V, -5 (A.20) and „-^fU 36.8 V. A.6 Matrices A system of simultaneous linear equations can also be solved using matrices. In what follows, we briefly review matrix notation, algebra, and terminology. 1 A matrix is by definition a rectangular array of elements; thus A = #ii a n a u (l 2 \ (l 22 & 2 T, _"ml "-ml "m3 «1» ttln (A.21) is a matrix with m rows and n columns. We describe A as being a matrix of order m by n, or m X «, where m equals the number of rows and n the 1 An excellent introductory-level text in matrix applications to circuit analysis is Lawrence P. Huelsman, Circuits, Matrices, and Linear Vector Spaces (New York: McGraw-Hill, 1963). number of columns. We always specify the rows first and the columns sec- ond. The elements of the matrix — « n , a 12 , «13, • • .—can be real numbers, complex numbers, or functions. We denote a matrix with a boldface capi- tal letter. The array in Eq. A.21 is frequently abbreviated by writing A [ a ij\mii > where a tj is the element in the /th row and theyth column. If m — 1, A is called a row matrix, that is, A « [flu a n a l3 ••• a h! ). If /2 = 1, A is called a column matrix, that is, (A.22) (A.23) «11 «21 A = a 3l . (A. 24) . a m\. If m = n, A is called a square matrix. For example, if m = n = 3, the square 3 by 3 matrix is A = «21 «22 «23 • (A.25) «11 «21 «31 «12 «22 «32 «13 «23 «33 Also note that we use brackets [] to denote a matrix, whereas we use vertical lines 11 to denote a determinant. It is important to know the differ- ence. A matrix is a rectangular array of elements. A determinant is a func- tion of a square array of elements. Thus if a matrix A is square, we can define the determinant of A. For example, if 2 1 6 15 then detA 30 - 6 = 24. A,7 Matrix Algebra The equality, addition, and subtraction of matrices apply only to matrices of the same order. Two matrices are equal if, and only if, their correspon- ding elements are equal. In other words, A = B if, and only if, a^ = b,j for all i and ;*. For example, the two matrices in Eqs. A.26 and A.27 arc equal because a n = b n ,a u = b 12 ,a 2 i = 6 2 i,and« 2 2 = ^22 : "36 4 -20 16 (A.26) B = 36 -20 4 16 If A and B are of the same order, then (A.27) C = A + B (A.28) implies For example, if ij ^U ij' (A.29) 4 -6 10 8 12 -4 (A.30) and B 16 10 -30 -20 8 15 (A.31) then 20 4 -20 -12 20 11 (A.32) The equation D = A - B (A.33) implies djj = a tj - bjj. For the matrices in Eqs. A.30 and A.31, we would have (A.34) D -12 -16 40 28 4 -19 (A.35) Matrices of the same order are said to be conformable for addition and subtraction. Multiplying a matrix by a scalar k is equivalent to multiplying each element by the scalar. Thus A = kB if, and only if, a- t ; = kbn. It should be noted that k may be real or complex. As an example, we will multiply the matrix D in Eq. A.35 by 5. The result is 5D -60 -80 200 140 20 -95 (A.36) Matrix multiplication can be performed only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In other words, the product AB requires the number of columns in A to equal the number of rows in B.The order of the resulting matrix will 712 The Solution of Linear Simultaneous Equations be the number of rows in A by the number of columns in B. Thus if C = AB, where A is of order m X p and B is of order /; x n, then C will be a matrix of order m X n. When the number of columns in A equals the number of rows in B, we say A is conformable to B for multiplication. An element in C is given by the formula (A.37) The formula given by Eq. A.37 is easy to use if one remembers that matrix multiplication is a row-by-column operation. Hence to get the /th, /th term in C, each element in the /th row of A is multiplied by the corre- sponding element in the /th column of B, and the resulting products are summed. The following example illustrates the procedure. We are asked to find the matrix C when A = 6 3 2 1 4 6 (A.38) and B ~4 0 _1 2~ 3 -2_ (A.39) First we note that C will be a 2 X 2 matrix and that each element in C will require summing three products. To find Cj! we multiply the corresponding elements in row 1 of matrix A with the elements in column 1 of matrix B and then sum the products. We can visualize this multiplication and summing process by extracting the corresponding row and column from each matrix and then lining them up element by element. So to find C u we have Row 1 of A Column 1 of B 6 4 3 0 2 1 therefore C n = 6 X 4 + 3 x 0 + 2 X 1 = 26. To find Cp we visualize Row 1 of A 6 Column 2 of B 2 -2 ' thus C 12 = 6X2 + 3X3 + 2X (-2) = 17. For C 2 \ we have Row 2 of A Column 1 of B 1 4 4 0 6 1 and C 2 \ = 1X4 + 4X0 + 6X1 = 10. Finally, for C 2 2 we have Row 2 of A 1 from which Column 2 of B 2 -2 ' C 22 = 1 X 2 + 4 X 3 + 6 X (-2) = 2. It follows that AB = 26 17 10 2 (A.40) In general, matrix multiplication is not commutative, that is, AB & BA. As an example, consider the product BA for the matrices in Eqs. A.38 and A.39. The matrix generated by this multiplication is of order 3X3, and each term in the resulting matrix requires adding two products. Therefore if D = BA, we have D = 26 3 4 20 12 -5 20 18 -10 (A.41) Obviously, C =£ D. We leave you to verify the elements in Eq. A.41. Matrix multiplication is associative and distributive. Thus (AB)C = A(BC), (A.42) and A(B + C) = AB + AC, (A + B)C = AC + BC. (A.43) (A.44) In Eqs. A.42, A.43, and A.44, we assume that the matrices are conformable for addition and multiplication. We have already noted that matrix multiplication is not commutative. There are two other properties of multiplication in scalar algebra that do not carry over to matrix algebra. First, the matrix product AB = 0 does not imply either A = 0 or B = 0. (Note: A matrix is equal to zero when all its elements are zero.) For example, if A = 1 0 2 0 and B = '0 0' 4 8 then AB "0 0 0 0 Hence the product is zero, but neither A nor B is zero. Second, the matrix equation AB = AC does not imply B = C. For example, if A = 1 0 2 0_ B = V, U 8. , and C = "3 4 .5 6 then AB = AC = 3 4 6 8 butB 56 C. The transpose of a matrix is formed by interchanging the rows and columns. For example, if A = 1 2 3 4 5 6 7 8 9 , then A r = 1 4 7 2 5 8 3 6 9 The transpose of the sum of two matrices is equal to the sum of the transposes, that is, T _ AT (A + B) y = A' + B'. (A.45) The transpose of the product of two matrices is equal to the product of the transposes taken in reverse order. In other words, [AB] 7 ' = B T A r . (A.46) Equation A.46 can be extended to a product of any number of matri- ces. For example, T _ nTr-TttTAT [ABCD]' = D'C'B'A (A.47) If A = A 7 , the matrix is said to be symmetric. Only square matrices can be symmetric. A.8 Identity, Adjoint, and Inverse Matrices An identity matrix is a square matrix where a (/ = 0 for i <£ y, and a i} = 1 for i = j. In other words, all the elements in an identity matrix are zero except those along the main diagonal, where they are equal to l.Thus and 1 0 0' 1, 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0~ 0 0 1_ are all identity matrices. Note that identity matrices are always square. We will use the symbol U for an identity matrix. The adjoint of a matrix A of order n X n is defined as adjA = [A/zkxa, (A.48) where A, y is the cofactor of %. (See Section A.5 for the definition of a cofactor.) It follows from Eq. A.48 that one can think of finding the adjoint of a square matrix as a two-step process. First construct a matrix made up of the cofactors of A, and then transpose the matrix of cofactors. As an example we will find the adjoint of the 3x3 matrix 1 3 1 2 2 1 3 1 5 The cofactors of the elements in A are A„= 1(10- 1) = 9, A 12 = -1(15 + 1) = -16, A ]3 = 1(3 + 2) = 5, A 21 = -1(10 - 3) = -7, A 22 = 1(5 + 3) = 8, A 23 = -1(1 +2) = -3, A 31 = 1(2 - 6) = -4, A 32 = -1(1 - 9) = 8, A 33 = 1(2 - 6) = -4. The matrix of cofactors is B 9 -7 -4 -16 S 8 5 -3 -4 It follows that the adjoint of A is adj A = B 7 = 9 -16 5 -7 8 -3 -4 8 -4 One can check the arithmetic of finding the adjoint of a matrix by using the theorem adj A • A = det A • U. (A.49) Equation A.49 tells us that the adjoint of A times A equals the determi- nant of A times the identity matrix, or for our example. det A = 1(9) + 3(-7) - 1(-4) = -8. . excellent introductory-level text in matrix applications to circuit analysis is Lawrence P. Huelsman, Circuits, Matrices, and Linear Vector Spaces (New York: McGraw-Hill, 1963). number of columns.