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The matrix of coefficients either have no solution or have infinite solutions is system of equations are
- a)Linearly dependent
- b)Linearly independent
- c)Quadratic and independent
- d)Quadratic and dependent
Correct answer is option 'A'. Can you explain this answer?
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The matrix of coefficients either have no solution or have infinite so...
Explanation:
When we talk about the matrix of coefficients, we are referring to the matrix that contains the coefficients of the variables in the system of equations. For example, in the system of equations:
2x + 3y = 5
4x + 6y = 10
The matrix of coefficients would be:
[2 3]
[4 6]
Now, if the matrix of coefficients either has no solution or has infinite solutions, it means that the system of equations is either linearly dependent or inconsistent.
Linearly Dependent:
A set of equations is linearly dependent if one or more of the equations can be expressed as a linear combination of the others. This means that one of the equations is redundant and can be eliminated without changing the solution. In this case, the matrix of coefficients will have a row of zeros or a row that is a multiple of another row. This will result in a system of equations that has infinite solutions, since we can choose any value for the eliminated variable.
Inconsistent:
A set of equations is inconsistent if there is no solution that satisfies all of the equations. This happens when the equations are contradictory, such as:
2x + 3y = 5
2x + 3y = 10
In this case, the matrix of coefficients will have a row of zeros, which means that one of the variables is not involved in any of the equations. This will result in a system of equations that has no solution.
Therefore, if the matrix of coefficients either has no solution or has infinite solutions, the system of equations is linearly dependent.
When we talk about the matrix of coefficients, we are referring to the matrix that contains the coefficients of the variables in the system of equations. For example, in the system of equations:
2x + 3y = 5
4x + 6y = 10
The matrix of coefficients would be:
[2 3]
[4 6]
Now, if the matrix of coefficients either has no solution or has infinite solutions, it means that the system of equations is either linearly dependent or inconsistent.
Linearly Dependent:
A set of equations is linearly dependent if one or more of the equations can be expressed as a linear combination of the others. This means that one of the equations is redundant and can be eliminated without changing the solution. In this case, the matrix of coefficients will have a row of zeros or a row that is a multiple of another row. This will result in a system of equations that has infinite solutions, since we can choose any value for the eliminated variable.
Inconsistent:
A set of equations is inconsistent if there is no solution that satisfies all of the equations. This happens when the equations are contradictory, such as:
2x + 3y = 5
2x + 3y = 10
In this case, the matrix of coefficients will have a row of zeros, which means that one of the variables is not involved in any of the equations. This will result in a system of equations that has no solution.
Therefore, if the matrix of coefficients either has no solution or has infinite solutions, the system of equations is linearly dependent.
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Community Answer
The matrix of coefficients either have no solution or have infinite so...
As due to linear dependency the rank of the matrix will be less than the number of unknown variables... so this lead to infinite solution or no solution....
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The matrix of coefficients either have no solution or have infinite solutions is system of equations area)Linearly dependentb)Linearly independentc)Quadratic and independentd)Quadratic and dependentCorrect answer is option 'A'. Can you explain this answer?
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