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Number of solutions

Linear systems & matrices

Japanese school year: University year 1

What you learn

Learn how to determine the number of solutions by comparing the ranks of the coefficient and augmented matrices with the number of unknowns. If the ranks are equal, the system has a unique solution when equal to the unknowns, or infinitely many when smaller. Understanding elementary row operations is required.

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Key points

This condition determines whether a system of linear equations has exactly one unique solution. It holds when the rank of coefficient matrix AA and augmented matrix [A∣b][A \mid \boldsymbol{b}] are equal, and both match the number of variables nn.

rank⁡(A)=rank⁡([A∣b])=n\operatorname{rank}(A) = \operatorname{rank}([A \mid \boldsymbol{b}]) = n

This condition shows that a linear system has infinitely many solutions. When the ranks of AA and [A∣b][A \mid \boldsymbol{b}] are equal but less than the number of variables nn, free variables appear and allow infinite solutions.

rank⁡(A)=rank⁡([A∣b])<n\operatorname{rank}(A) = \operatorname{rank}([A \mid \boldsymbol{b}]) < n

This condition tests whether a system of linear equations has no solution at all. When the augmented matrix [A∣b][A \mid \boldsymbol{b}] has a strictly higher rank than coefficient matrix AA, a contradiction occurs and no solution can exist.

rank⁡(A)<rank⁡([A∣b])\operatorname{rank}(A) < \operatorname{rank}([A \mid \boldsymbol{b}])

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