Home > Algebra calculators > Linear Equation in three variables using Inverse matrix method, Gauss elimination method calculator

1. Find Inverse of matrix A
A = OR A =

 2. Solve Simultaneous Equations Enter Equations with ',' seperated: OR Enter Equations line by line: 2x+y+z=5 3x+5y+2z=15 2x+y+4z=8
SolutionHelp1.1 Inverse1.2 Inverse Guess Elimination2.1 Inverse Matrix2.2 Cramer's Rule2.3 Gauss Elimination2.4 Gauss Sield
 2. Solve Simultaneous Equations using Inverse Matrix method 2. Solve Equations 2x+y+z=5,3x+5y+2z=15,2x+y+4z=8 using Inverse Matrix methodHere 2x+y+z=5, 3x+5y+2z=15, 2x+y+4z=8Now converting given equations into matrix form [[2,1,1],[3,5,2],[2,1,4]] [[ x ],[ y ],[ z ]]=[[5],[15],[8]]Now, A = [[2,1,1],[3,5,2],[2,1,4]], X = [[ x ],[ y ],[ z ]] and B = [[5],[15],[8]]:. AX = B:. X = A^-1 B| A |=|[2,1,1],[3,5,2],[2,1,4]| = 2 (5 × 4 - 2 × 1) - 1 (3 × 4 - 2 × 2) + 1 (3 × 1 - 5 × 2) = 2 (20 - 2) - 1 (12 - 4) + 1 (3 - 10) = 2 (18) - 1 (8) + 1 (-7) = 36 - 8 - 7 = 21"Here, " | A | = 21 != 0 :. A^(-1)" is possible."Adj(A)=Adj[[2,1,1],[3,5,2],[2,1,4]]=[[+(5 × 4 - 2 × 1),-(3 × 4 - 2 × 2),+(3 × 1 - 5 × 2)],[-(1 × 4 - 1 × 1),+(2 × 4 - 1 × 2),-(2 × 1 - 1 × 2)],[+(1 × 2 - 1 × 5),-(2 × 2 - 1 × 3),+(2 × 5 - 1 × 3)]]^T=[[18,-8,-7],[-3,6,0],[-3,-1,7]]^T=[[18,-3,-3],[-8,6,-1],[-7,0,7]]"Now, "A^(-1)=1/| A | × Adj(A)"Here, "X = A^(-1) × B:. X =1/| A | × Adj(A) × B=1/21 × [[18,-3,-3],[-8,6,-1],[-7,0,7]] × [[5],[15],[8]] =1/21 ×[[18*5 + -3*15 + -3*8],[-8*5 + 6*15 + -1*8],[-7*5 + 0*15 + 7*8]] =1/21 ×[[21],[42],[21]] =[[1],[2],[1]]:.[[ x ],[ y ],[ z ]]=[[1],[2],[1]]:. x=1, y=2, z=1