If the augmented matrix does not tell us there is no solution and if there is no free variable (i.e. every column other than the right-most column is a pivot column), then the system has a unique solution. For example, if A= and b=, then there is a unique solution to the system Ax=b.
Condition for Unique Solution to Linear Equations A system of linear equations ax + by + c = 0 and dx + ey + g = 0 will have a unique solution if the two lines represented by the equations ax + by + c = 0 and dx + ey + g = 0 intersect at a point. i.e., if the two lines are neither parallel nor coincident.
Note: To know about the infinite solution of a matrix first we have to check nonzero rows in the matrix. That means if the number of variables is more than nonzero rows then that matrix has an infinite solution.
(i) Consistent equations with unique solution: The graphs of two equations intersect at a unique point. For Example Consider. x + 2y = 4. 7x + 4y = 18. The graphs (lines) of these equations intersect each other at the point (2, 1) i.e., x = 2, y = 1.
In a set of linear simultaneous equations, a unique solution exists if and only if, (a) the number of unknowns and the number of equations are equal, (b) all equations are consistent, and (c) there is no linear dependence between any two or more equations, that is, all equations are independent.
As you can see, the final row of the row reduced matrix consists of 0. This means that for any value of Z, there will be a unique solution of x and y, therefore this system of linear equations has infinite solutions.
A system of linear equations has infinite solutions when the graphs are the exact same line.
We are not guaranteed to have one and only one solution, which in many cases is because we have more equations than unknowns (m bigger n).
Hint. That the inverse matrix of A is unique means that there is only one inverse matrix of A. (That's why we say “the” inverse matrix of A and denote it by A−1.) So to prove the uniqueness, suppose that you have two inverse matrices B and C and show that in fact B=C.
Answer: When a pair of linear equations has a unique solution lines are Intersecting.
Theorem 1.2 provides the answer. Corollary 1.3 Let A be an m × n matrix. A homogeneous system of equations Ax = 0 will have a unique solution, the trivial solution x = 0, if and only if rank[A] = n. In all other cases, it will have infinitely many solutions.
If A is invertible, then its inverse is unique. Remark When A is invertible, we denote its inverse as A−1. Theorem. If A is an n × n invertible matrix, then the system of linear equations given by A x = b has the unique solution x = A−1b.
three wishesShenron When Dende became the Earth's guardian, he modified the burnt out Dragon Balls created by Kami, giving Shenron the power to grant three wishes (two wishes if one wish is used to revive a large amount of people).
Divorced or widowed men can be “sealed” (married for eternity in Latter-day Saint temples) to multiple wives, while such women generally can be sealed only to one husband.