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1. Find Inverse of matrix A
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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 Cramer's Rule method
2. Solve Equations 2x+y+z=5,3x+5y+2z=15,2x+y+4z=8 using Cramer's Rule method

The equations can be expressed as
`2x+y+z-5=0`

`3x+5y+2z-15=0`

`2x+y+4z-8=0`

Use Cramer’s Rule to find the values of x, y, z.
`(x)/D_x=(-y)/D_y=(z)/D_z=(-1)/D`

` D_x =|[1,1,-5],[5,2,-15],[1,4,-8]|`

` = 1 (2 × -8 - -15 × 4) - 1 (5 × -8 - -15 × 1) + (-5) (5 × 4 - 2 × 1)`

` = 1 (-16 - -60) - 1 (-40 - -15) - 5 (20 - 2)`

` = 1 (44) - 1 (-25) - 5 (18)`

` = 44 + 25 - 90`

` = -21`


` D_y =|[2,1,-5],[3,2,-15],[2,4,-8]|`

` = 2 (2 × -8 - -15 × 4) - 1 (3 × -8 - -15 × 2) + (-5) (3 × 4 - 2 × 2)`

` = 2 (-16 - -60) - 1 (-24 - -30) - 5 (12 - 4)`

` = 2 (44) - 1 (6) - 5 (8)`

` = 88 - 6 - 40`

` = 42`


` D_z =|[2,1,-5],[3,5,-15],[2,1,-8]|`

` = 2 (5 × -8 - -15 × 1) - 1 (3 × -8 - -15 × 2) + (-5) (3 × 1 - 5 × 2)`

` = 2 (-40 - -15) - 1 (-24 - -30) - 5 (3 - 10)`

` = 2 (-25) - 1 (6) - 5 (-7)`

` = -50 - 6 + 35`

` = -21`


` D =|[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`


`(x)/D_x=(-y)/D_y=(z)/D_z=(-1)/D`

`:. (x)/-21=(-y)/42=(z)/-21=(-1)/21`

`:. (x)/-21=(-1)/21,(-y)/42=(-1)/21,(z)/-21=(-1)/21`

`:. x=(21)/(21),y=(42)/(21),z=(21)/(21)`

`:. x=1,y=2,z=1`

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