Inverse of a 3x3 Matrix and Solving a System of Linear Equations by Gauss Elimination

Published on November 2, 2026

Linear-Algebra

Find the inverse of the 3x3 matrix with rows (2,1,1), (1,3,2), (1,0,2) using the adjugate and determinant method, then solve the system 8x+y+z=10, 2x+9y+z=12, x+y+7z=9 with Gauss elimination and back substitution to get x=1, y=1, z=1.

Question

Find the inverse of the following matrix: [■(2&1&1@1&3&2@1&0&2)] Use the gauss elimination method to solve the following system: 8x + y + z =10 2x + 9y + z =12 x + y +7z = 9

Answer

Part 1: Inverse of the Matrix

Let

Determinant:

Cofactors:

Adjugate (transpose of the cofactor matrix):

Inverse:

Part 2: Solving the System by Gauss Elimination

Forward elimination on :

Multiplying these by 4 and 8 respectively gives:

Eliminate :

Back substitution:

Check:

✓

Final Answers

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