## Quick Reference

A set of *m* linear equations in *n* unknowns *x*_{1}, *x*_{2},…, *x*_{n} that has the form

*a*_{11}*x*_{1} + *a*_{12}*x*_{2} + ⋯ + *a*_{1n}*x*_{n} = *b*_{1},

*a*_{21}*x*_{1} + *a*_{22}*x*_{2} + ⋯ + *a*_{2n}*x*_{n} = *b*_{2},

⋮

*a*_{m1}*x*_{1} + *a*_{m2}*x*_{2} + ⋯ + *a*_{mn}*x*_{n} = *b*_{m},

where *b*_{1}, *b*_{2},…, *b*_{m} are not all zero. (Compare with homogeneous set of linear equations.) In matrix notation, this set of equations may be written as **Ax**=**b**, where **A** is the *m*×*n* matrix [*a*_{ij}], and **b**(≠**0**) and **x** are column matrices:

Such a set of equations may be inconsistent, have a unique solution, or have infinitely many solutions (see simultaneous linear equations). For a set consisting of the same number of linear equations as unknowns, the matrix of coefficients **A** is a square matrix, and the set of equations has a unique solution, namely **x**=**A**^{−1}**b**, if and only if **A** is invertible.

*Subjects:*
Mathematics.

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