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In the matrix equation Px = q, which of the following is a necessary condition for the existence of at least one solution for the unknown vector x?
- a)Matrix P must be square.
- b)Vector q must have only non-zero element.
- c)Matrix P must be singular.
- d)Augmented matrix [P q] must have the same rank as matrix P.
Correct answer is option 'D'. Can you explain this answer?
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In the matrix equation Px = q, which of the following is a necessary ...
Explanation:
To understand the necessary condition for the existence of at least one solution for the unknown vector x in the matrix equation Px = q, let's consider each option:
a) Matrix P must be square:
This condition is not necessary for the existence of at least one solution. A non-square matrix can still have a solution if the number of rows in P is greater than or equal to the number of columns.
b) Vector q must have only non-zero element:
This condition is not necessary for the existence of at least one solution. The vector q can have zero elements, and the matrix equation can still have a solution.
c) Matrix P must be singular:
This condition is not necessary for the existence of at least one solution. A non-singular matrix can still have a solution if the vector q lies in the column space of the matrix P.
d) Augmented matrix [P q] must have the same rank as matrix P:
This condition is the necessary condition for the existence of at least one solution. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. If the augmented matrix [P q] has the same rank as matrix P, it means that the vector q lies in the column space of matrix P. In other words, q can be expressed as a linear combination of the columns of P. This ensures that there exists at least one solution for the unknown vector x.
Therefore, the correct answer is option 'D' - the augmented matrix [P q] must have the same rank as matrix P. This condition is necessary for the existence of at least one solution for the unknown vector x in the matrix equation Px = q.
To understand the necessary condition for the existence of at least one solution for the unknown vector x in the matrix equation Px = q, let's consider each option:
a) Matrix P must be square:
This condition is not necessary for the existence of at least one solution. A non-square matrix can still have a solution if the number of rows in P is greater than or equal to the number of columns.
b) Vector q must have only non-zero element:
This condition is not necessary for the existence of at least one solution. The vector q can have zero elements, and the matrix equation can still have a solution.
c) Matrix P must be singular:
This condition is not necessary for the existence of at least one solution. A non-singular matrix can still have a solution if the vector q lies in the column space of the matrix P.
d) Augmented matrix [P q] must have the same rank as matrix P:
This condition is the necessary condition for the existence of at least one solution. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. If the augmented matrix [P q] has the same rank as matrix P, it means that the vector q lies in the column space of matrix P. In other words, q can be expressed as a linear combination of the columns of P. This ensures that there exists at least one solution for the unknown vector x.
Therefore, the correct answer is option 'D' - the augmented matrix [P q] must have the same rank as matrix P. This condition is necessary for the existence of at least one solution for the unknown vector x in the matrix equation Px = q.
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In the matrix equation Px = q, which of the following is a necessary ...
According to Rouche's theorem, the system is consistent if and only if the co-efficient matrix and the augmented matrix are of the same rank, otherwise the system is inconsistent.
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In the matrix equation Px = q, which of the following is a necessary condition for the existence of at least one solution for the unknown vector x?a)Matrix P must be square.b)Vector q must have only non-zero element.c)Matrix P must be singular.d)Augmented matrix [P q] must have the same rank as matrix P.Correct answer is option 'D'. Can you explain this answer?
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