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**Existence and Uniqueness of Matrices (and singular matrices)

Definition

An \(n \times n\) matrix \(\mathbf{A}\) is said to be non-singular if it satisfies any of the following equivalent conditions:

  1. \(\mathbf{A}\) has an inverse. (ie. there exists a matrix \(\mathbf{A}^{-1}\) such that \(\mathbf{AA^{-1} = A^{-1}A = I}\)).
  2. \(\text{det}(\mathbf{A}) \neq 0\); the determinant is non-zero.
  3. \(\mathbf{A}\) has rank \(n\). (the rank of the matrix is the number of linearly independent rows/cols it has).
  4. For any vector \(\mathbf{x} \neq \mathbf{0}\), \(\mathbf{Ax} \neq \mathbf{0}\). (matrix \(\mathbf{A}\) annihilates no nontrivial vector). Otherwise, the matrix is singular.

The existence and uniqueness of a solution to a system of linear equations \(\mathbf{Ax = b}\) is dependant on whether \(\mathbf{A}\) is non-singular or singular.

If the matrix is non-singular it has an inverse, and the system \(\mathbf{Ax=b}\) always has the unique solution \(\mathbf{x = A^{-1}b}\) regardless of the value for \(\mathbf{b}\).

If the matrix is singular, then the number of solutions is determined by : for a given value of there may be no solution, but if there is a solution \(\mathbf{x}\), so that \(\mathbf{Ax=b}\), then we also have \(\mathbf{A}(\mathbf{x} - \gamma \mathbf{z}) = \mathbf{b}\) for any scalar \(\gamma\), where \(\mathbf{z \neq 0}\) is a vector such that \(\mathbf{Az = 0}\). (Note that such a \(\mathbf{z}\) must exist, since otherwise Condition 4 implies that \(\mathbf{A}\) is nonsingular).

For a given matrix \(\mathbf{A}\) and a right-hand-side vector \(\mathbf{b}\), the posibilities are:

  1. Unique solution: \(\mathbf{A}\) nonsingular, \(\mathbf{b}\) arbitrary
  2. Infinitely many solutions: \(\mathbf{A}\) singular, \(\mathbf{b} \in \text{span}(\mathbf{A})\) (consistent)
  3. No solution: \(\mathbf{A}\) singular, \(\mathbf{b} \not\in \text{span}(\mathbf{A})\).

In two dimensions, each linear system represents a line on the plane:

  • a unique solution is the intersection of the two lines
  • an infinitely many solutions is when the two lines are parallel and overlap
  • no solution is when the two lines are parallel and don't overlap

In higher dimensions, each equation determines a hyperplane. In the nonsingular case, the unique solution is given by the intersection of all hyperplanes.