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  1. 23×33\times 3 determinant expansion
  2. 10Eigenvalues of a 3×33\times 3 matrix
  3. 12Eigenvector for eigenvalue 3
  4. 14Inverse of a 2×22\times 2 matrix
  5. 17Solving a 3×33\times 3 linear system
  6. 20System with no solution
  7. 21Singular matrix, det=0
  8. 22Eigenvalue zero of a matrix
  9. 27Null space of a matrix
  10. 37Volume via scalar triple product
  11. 43Characteristic polynomial, 2×22\times 2
  12. 47Orthogonal matrix check
  13. 48Gram-Schmidt second vector
  14. 49Least-squares line fit
  15. 50Coordinates in a new basis
  16. 51Distance point to plane

17Solving a 3×33\times 3 linear system

Solve the system {x+y+z=62y+5z=−42x+5y−z=27\displaystyle \begin{cases}x + y + z = 6\\2y + 5z = -4\\2x + 5y - z = 27\end{cases}

Eliminate x using equation 1, then solve the remaining pair in y and z. The solution is x = 5, y = 3, z = −2. Check: 2(5) + 5(3) − (−2) = 27 ✓.

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