Determine If Matrix Is Inconsistent

RS Aggarwal

In mathematics and specially in algebra, a linear or nonlinear system of equations is called consistent if there is at least 1 set of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation concord truthful as an identity. In contrast, a linear or not linear equation system is called inconsistent if there is no set of values for the unknowns that satisfies all of the equations.^{[1] [two]}

If a organization of equations is inconsistent, then it is possible to dispense and combine the equations in such a way every bit to obtain contradictory data, such as ii = i, or x ³ + y ^three = 5 and x ³ + y ⁱⁱⁱ = vi (which implies 5 = 6).

Both types of equation system, consistent and inconsistent, can exist any of overdetermined (having more equations than unknowns), underdetermined (having fewer equations than unknowns), or exactly determined.

Unproblematic examples [edit]

Underdetermined and consistent [edit]

The system

${\displaystyle x+y+z=3,}$

${\displaystyle x+y+2z=4}$

has an infinite number of solutions, all of them having z = 1 (as can be seen by subtracting the first equation from the second), and all of them therefore having x+y = two for any values of x and y.

The nonlinear organisation

${\displaystyle x^{ii}+y^{2}+z^{2}=10,}$

${\displaystyle x^{2}+y^{two}=5}$

has an infinitude of solutions, all involving ${\displaystyle z=\pm {\sqrt {5}}.}$

Since each of these systems has more than i solution, it is an indeterminate system.

Underdetermined and inconsistent [edit]

The system

${\displaystyle ten+y+z=3,}$

${\displaystyle x+y+z=4}$

has no solutions, as can be seen past subtracting the first equation from the 2d to obtain the impossible 0 = 1.

The not-linear organization

${\displaystyle 10^{2}+y^{ii}+z^{2}=17,}$

${\displaystyle ten^{2}+y^{2}+z^{two}=14}$

has no solutions, because if one equation is subtracted from the other we obtain the impossible 0 = 3.

Exactly determined and consistent [edit]

The system

${\displaystyle x+y=iii,}$

${\displaystyle x+2y=five}$

has exactly one solution: ten = 1, y = two.

The nonlinear organisation

${\displaystyle ten+y=1,}$

${\displaystyle 10^{2}+y^{2}=1}$

has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while

${\displaystyle x^{3}+y^{3}+z^{three}=10,}$

${\displaystyle x^{iii}+2y^{3}+z^{3}=12,}$

${\displaystyle 3x^{3}+5y^{3}+3z^{3}=34}$

has an space number of solutions because the third equation is the start equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y tin can be institute to satisfy the outset ii (and hence the third) equations.

Exactly determined and inconsistent [edit]

The system

${\displaystyle x+y=three,}$

${\displaystyle 4x+4y=ten}$

has no solutions; the inconsistency can exist seen by multiplying the first equation by 4 and subtracting the 2d equation to obtain the incommunicable 0 = 2.

Also,

${\displaystyle x^{3}+y^{three}+z^{iii}=10,}$

${\displaystyle 10^{3}+2y^{3}+z^{3}=12,}$

${\displaystyle 3x^{3}+5y^{3}+3z^{3}=32}$

is an inconsistent arrangement because the commencement equation plus twice the second minus the tertiary contains the contradiction 0 = 2.

Overdetermined and consistent [edit]

The system

${\displaystyle x+y=iii,}$

${\displaystyle 10+2y=7,}$

${\displaystyle 4x+6y=20}$

has a solution, ten = –one, y = four, because the first two equations practice not contradict each other and the tertiary equation is redundant (since it contains the same information as can be obtained from the first two equations past multiplying each through by 2 and summing them).

The arrangement

${\displaystyle 10+2y=7,}$

${\displaystyle 3x+6y=21,}$

${\displaystyle 7x+14y=49}$

has an infinitude of solutions since all three equations give the same information as each other (equally can be seen by multiplying through the beginning equation by either three or 7). Whatsoever value of y is role of a solution, with the corresponding value of x beingness 7–2y.

The nonlinear system

${\displaystyle x^{two}-1=0,}$

${\displaystyle y^{2}-1=0,}$

${\displaystyle (x-1)(y-ane)=0}$

has the three solutions (ten, y) = (1, –1), (–1, ane), and (1, one).

Overdetermined and inconsistent [edit]

The system

${\displaystyle x+y=iii,}$

${\displaystyle x+2y=vii,}$

${\displaystyle 4x+6y=21}$

is inconsistent considering the concluding equation contradicts the information embedded in the offset two, every bit seen by multiplying each of the get-go two through by 2 and summing them.

The system

${\displaystyle x^{2}+y^{ii}=1,}$

${\displaystyle x^{two}+2y^{two}=ii,}$

${\displaystyle 2x^{ii}+3y^{2}=4}$

is inconsistent because the sum of the first ii equations contradicts the third one.

Criteria for consistency [edit]

Every bit can be seen from the above examples, consistency versus inconsistency is a dissimilar event from comparing the numbers of equations and unknowns.

Linear systems [edit]

A linear organisation is consequent if and only if its coefficient matrix has the same rank as does its augmented matrix (the coefficient matrix with an actress cavalcade added, that column being the column vector of constants).

Nonlinear systems [edit]

References [edit]

1. ^ "Definition of CONSISTENT EQUATIONS". www.merriam-webster.com . Retrieved 2021-06-x .
2. ^ "Definition of consistent equations | Dictionary.com". www.dictionary.com . Retrieved 2021-06-10 .

Determine If Matrix Is Inconsistent,

Source: https://en.wikipedia.org/wiki/Consistent_and_inconsistent_equations

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