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6502 Hand Out 11

The document summarizes matrix-vector systems of linear equations. There are three possibilities for the number of solutions to the system Ax = b: a unique solution, an infinite number of solutions, or no solutions. The determinant of the matrix A determines whether there is a unique solution or not. If the determinant is non-zero, there is a unique solution. If the determinant is zero, there may be no solution or an infinite number of solutions. The document also distinguishes between homogeneous systems where b = 0, and inhomogeneous systems where b ≠ 0.

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Roy Vesey
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50 views1 page

6502 Hand Out 11

The document summarizes matrix-vector systems of linear equations. There are three possibilities for the number of solutions to the system Ax = b: a unique solution, an infinite number of solutions, or no solutions. The determinant of the matrix A determines whether there is a unique solution or not. If the determinant is non-zero, there is a unique solution. If the determinant is zero, there may be no solution or an infinite number of solutions. The document also distinguishes between homogeneous systems where b = 0, and inhomogeneous systems where b ≠ 0.

Uploaded by

Roy Vesey
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Handout 11 Matrix-vector systems

We consider a system of n equations in n unknowns, which can be written in matrix-vector form as

Ax = b

where A is an n × n matrix.

How many solutions?

There are three possibilities:

• Unique solution
• Infinite number of solutions

• No solutions

Geometrically we can see these in 3D, where an equation represents a plane, and a solution means a
point lying on three planes:

¡@ @¡
¡ ¡ ¡
¡ ¡ ¡ ¡@@
@ ¡ @
¡ @@ @ ¡ @¡ ¡¡
¡ ¡
@ @ ¡¡ @
q
¡ ¡ ¡
¡ ¡ ¡ ¡ @ ¡¡¡¡@ ¡¡ ¡
¡@ ¡
¡¡ ¡ @¡
Unique solution Infinite solutions No solutions

Conditions for a unique solution

Whether or not there is a unique solution depends on the determinant of A:

• det (A) 6= 0 means there is a unique solution to A x = b.


• If det (A) = 0 there may be no solutions to the equation, or there may be an infinite number of
them.

Homogeneous and inhomogeneous systems

An inhomogeneous system is one where b 6= 0. Typically we will be looking for a unique solution so
we will need det (A) 6= 0.
A homogeneous system is one where b = 0. We know one solution: x = 0, the trivial solution.
Typically we will be looking for a non-trivial solution: so we need more than one solution. That means
we need an infinite number of solutions and we will need det (A) = 0 to get them.

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