silemus.
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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

20System with no solution

For which value of a\displaystyle a does the system {x+y=2x+ay=3\displaystyle \begin{cases}x + y = 2\\x + ay = 3\end{cases} have no solution?

Subtracting the equations gives (a−1)y = 1, which is unsolvable only when a−1 = 0, so a = 1. Check: at a = 1 both equations become x+y = 2 and x+y = 3, parallel lines that never meet.

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