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1. Find Inverse of matrix A
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2. Solve Simultaneous Equations
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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 method

Here `2x+y+z=5`, `3x+5y+2z=15`, `2x+y+4z=8`

Now 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`

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