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DQ " 'I:oeYxNY?[vs+EEڦ[$i } `CJ4crVZ̤[qx2'Ob?,68nؘ9q_\Q>vvZB$w@G 4 gUT`J` Ab|Ûa!=էabzK3'i3B"g ZvSz@$Gn-5#%3Vg:dTlp9&I;Yg%`EԇI_\Y\ܭhފPH))Zt|>.@D Wc:͘D(K.gÌ#REЂl ))9fO"z!,9Esb"̤"iTxɟA?#AabS 2Qqb*Z[ q_ B$6)$AŅ"Y>Zkz'KNNt R;qA-TQؓ(W ,ANRɑLW"D3l:-`\,DY P4g dgm-6%[ P&zҝbK(So5#ڬ3hKE1+Eҝbk,j4/PM^˖e)1n&п%lZf8W*0,S; ,YJgZ._zԧ^&24Z$u8mT!3lbC.<:}/^ݗaIj%*;j%OF??~skIENDB`$b^u)̀ex%n^u)̀ex%PNG  IHDRl3~gAMA|Q pHYs+tEXtSoftwareMicrosoft Office5q(IDATc @_?7?0'#cE<IENDB`3 A@@  HABProduct of 2 Matrices [A][B] is =MMULT(array1,array2)9The matrix product array is the array of sums of product.5The cell location states which column and row to use.The 3rows of the first must match in dimension with the columns of the second %or the COUNTER will not be the same. Each cell is the Sum(Ri*Ci) )is =MINVERSE(A2:B3) Inverse of the matrixLinear System in Three Unknowns 2007, A. Azzolinoaugmented arrayAx+By+Cz = DCEx+Fy+GzHEFGIx+Jy+KzLIJKdiagonal productsSUMS determinantsolutionx = *y =z= x isy is z isSolve system by Cramer's Rule[A][B] *  is []1x33x11x12x32x12x23x3plusRow1Row2k1k2rowArowB (k1)(rowA) (k2)(rowB)(k1)(rowA)+(k2)(rowB)Row31.Original matrix.2.Identity matrix4.Compete computation.)5.Record the results here & recopy below. R2+R1-->R11,11,21,31,41,52,12,22,32,42,53,13,23,33,43,54,14,24,34,44,55,15,25,35,45,5[X] is =MMULT(,)Think ROW times COLUMN.AThink ADD the PRODUCTS of the ROW and the matching COLUMN entry. Cell name matches row & column.Multiply 2 Matrices.&Think ADD the PRODUCTS of the ROW and the matching COLUMN entry. %+!Matrices must match in dimension!Add corresponding cells. - examples/The sum matrix is the sums corresponding cells.3The COLUMN Dimension of the first factor must EQUAL(the ROW dimension of the second factor. m by nn by pm by p3If required column and row dimensions do not match,multiplication is not defined. The product matrix dimension is A(first factor ROW dimension) by (second factor COLUMN dimension).()([[]]] * [] = ))])]]3x2)+(1x2 = [ = [[.Name _________________________________________3mathnstuff.com/math/algebra/qb/mayhs2/a2p12.inverse 2007, A. Azzolino<In algebra I, to solve Ax = B, divide both sides by A to getx= B/AMIn algebra II, to solve [A][X]=[B], multiply both sides by the inverse of [A][X]=inv([A])([B])(Your task is to find the inverse of [A].RYour task is to find the matrix that multiplied by [A] yields the identity matrix.XFirst: Multiply the [A] matrix by its inverse matrix [I] listing the 2x2 product matrix.RSecond: Set each of the cells of the prouct matrix equal to its corresponding cellof the identity matrix = 1 = 0[I] =CFourth: Restate in inverse in terms of |A|, the determinant of [A].4Third: Consider each column a system of 2 equations.4 Solve the system using the values A, B, C, and D.Show work on back of sheet.1Fifth: Note: If |A| = 0, then [A] has no inverse.FIf A is [[A B][C D]], then the inverse of [A] is 1/|A|([[D -B][-C A]]. AI11+BI21 CI11+DI21 AI12+BI22 CI11+DI22CI11+AI11+BI21DI211D-0BAD-BC0A-1C D/(AD-BC) -C/(AD-BC)AI12+CI12+ CI12+DI22BI22DI220D-1BAD-CB1A-0C -B/(AD-BC) A/(AD-BC)D/|A| -C/|A| - B/|A|A/|A||A| =  - C - B AA =Solve the System with Matricesx21] * [[]] [x11] = [[]] [ inv([A]) =det([A]) = |A| =inv([A]) R1-R2-->R1 R1-R2-->R2 4R2+R1-->R1 0R2-R1-->R2[A] =Row4+Solve a System by Gauss-Jordan Elimination.01.Record original coefficient & constant matrix. =3diagonal and 0s elsewhere in the coefficient array..2.Compete computation to place 1s on the major3.Read solution.3 by 3 inverseE H A B C INV([A])=(1/|A|)*Inverse, Determinant 2003, 2008, A. Azzolino<Follow instructions on page. Many cells are self-computing.PageContentstitle & contents maytrix.xls add subtractmultiplymultiply colored r,cinverserow transfrom to find inversesolve system by inversesolve system, Gauss-Jordan elimEnter data in yellow cells.!Notes and worksheet follow below.:Compute inverse of 2x2 or 3x3. Contains notes & worksheet.7Enter data in yellow elements. The result is in green.+Enter data in upper right element location.TAdd 1x3, 2x3, 2x2, 3x3, or up to 5x5 arrays using matrix rules. Subtract up to 5x5.UMultiply a 1x3 times a 3x1, a 2x3 times a 3x1, a 2x2 times a 2x2, a 3x3 times a 3x3, %and conformable m by n times a n by pEnter data below ONLY!<DO NOT, DO NOT, DO NOT enter data below the red row!!!!!!!!!9The color-coded cells indicate the source of each factor.<Multiply a 1x3 by a 3x2, a 2x3 by a 3x2, and a 2x3 by a 3x3 4with color-coding to show the origin of each factor.introinverse answer keyCAnswer key to worksheet and notes on computation of 3 by 3 inverse.)Cells are color coded for identification.This page is not hot.1 The goal is to find the inverse of a matrix by /row manipulation and verify it by spread sheet.EBegin with a 2x2 or 3x3 matrix (entered in upper right corner at 1.).<An appropriate identity matrix is provided in green (at 2.).a11a12a13a21a31a22a23a32a33i11i12i13i21i22i31i32i33a11i11+a12i21+a13i31a11i21+a12i22+a13i32a11i13+a12i23+a13i33i23a21i11+a22i21+a23i31a31i11+a32i21+a33i31a21i12+a22i22+a23i32a31i12+a32i22+a33i32a21i13Ia22i23+a23i33a31i13+a32i23+a33i33'Find matrix I, the inverse of matrix A.Complete the multiplication.,State the systems of equations to be solved.,Restate the systems so varialbles are clear.,This system has variables i11, i21, and i31.(This system has variables i12, i22, i32.a11i12+a12i22+a13i32,This system has variables i13, i23, and i33.0To solve for i11, i21, and i31, use this system.0To solve for i13, i23, and i33, use this system.2Use Cramer's rule to sollve for i11, i21, and i31.1Use Cramer"s Rule to solve for i13, i23, and i33.0To solve for i12, i22, and i32, use this system.1Use Cramer's Rule to solve for i12, i22, and i32.D BThe goal is to change the element to the desired Identity element.3The user must decide what row operation to perform.7The spread sheet will complete the desired computation.GThe user must recopy the edited array and repeat the process as needed.ENote that the row operation notation is part of the answer. Save it.is =MMULT(F19:H19,K19:K21)] [x31Hot cells. Enter system here.matrix vs. cramers 3x33x3 inverse derivation<UNDER CONSTRUCTION See subscripted solution to solving a 3x3BUser & Spread sheet jointly use row transformation to find inverse6Hot 3x3 solution of system by matrix and Cramer's Rule9UNDER CONSTRUCTION Solve system, Gauss-Jordan elimination!Solve system by inverse, 2x2, 3x3"Scrap work" to inv[A]4Compute the Inverse of a Matrix by row manipulation.Choose a 2x2 or a 3x3. 6.Verify by formula. 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