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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)Augmented matrix [Pq] must have the same rank as matrix P
- b)Vector q must have only non-zero elements
- c)Matrix P must be singular
- d)Matrix P must be square
Correct answer is option 'A'. Can you explain this answer?
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In the matrix equation Px = q which of the following is a necessary co...
Explanation:
To understand why option A is the correct answer, let's first define the terms used in the question:
- P: The coefficient matrix with dimensions m x n (m rows and n columns)
- x: The unknown vector with dimensions n x 1 (n rows and 1 column)
- q: The constant vector with dimensions m x 1 (m rows and 1 column)
The matrix equation Px = q represents a system of linear equations. Each row of the coefficient matrix P, when multiplied by the vector x, should equal the corresponding element in the constant vector q.
In order for a solution to exist for the unknown vector x, the following condition must be satisfied:
A solution exists if and only if the rank of the augmented matrix [P|q] is equal to the rank of matrix P.
Explanation of Option A:
Option A states that the augmented matrix [Pq] must have the same rank as matrix P.
- The augmented matrix [Pq] is obtained by appending the constant vector q as an additional column to the coefficient matrix P.
- The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix.
If the augmented matrix [Pq] has the same rank as matrix P, it means that the additional column q does not introduce any new linearly independent rows or columns. This implies that the system of linear equations represented by the matrix equation Px = q has a consistent solution.
Explanation of Other Options:
- Option B: Vector q having only non-zero elements does not guarantee the existence of a solution. It only ensures that the constant vector q is not a zero vector.
- Option C: Matrix P being singular (i.e., not invertible) does not guarantee the existence of a solution. A singular matrix can have either infinitely many solutions or no solution at all, depending on the specific system of linear equations.
- Option D: Matrix P does not need to be square for a solution to exist. The dimensions of matrix P, vector x, and vector q are independent of each other.
In conclusion, the correct answer is option A. For a solution to exist for the unknown vector x in the matrix equation Px = q, the augmented matrix [Pq] must have the same rank as matrix P.
To understand why option A is the correct answer, let's first define the terms used in the question:
- P: The coefficient matrix with dimensions m x n (m rows and n columns)
- x: The unknown vector with dimensions n x 1 (n rows and 1 column)
- q: The constant vector with dimensions m x 1 (m rows and 1 column)
The matrix equation Px = q represents a system of linear equations. Each row of the coefficient matrix P, when multiplied by the vector x, should equal the corresponding element in the constant vector q.
In order for a solution to exist for the unknown vector x, the following condition must be satisfied:
A solution exists if and only if the rank of the augmented matrix [P|q] is equal to the rank of matrix P.
Explanation of Option A:
Option A states that the augmented matrix [Pq] must have the same rank as matrix P.
- The augmented matrix [Pq] is obtained by appending the constant vector q as an additional column to the coefficient matrix P.
- The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix.
If the augmented matrix [Pq] has the same rank as matrix P, it means that the additional column q does not introduce any new linearly independent rows or columns. This implies that the system of linear equations represented by the matrix equation Px = q has a consistent solution.
Explanation of Other Options:
- Option B: Vector q having only non-zero elements does not guarantee the existence of a solution. It only ensures that the constant vector q is not a zero vector.
- Option C: Matrix P being singular (i.e., not invertible) does not guarantee the existence of a solution. A singular matrix can have either infinitely many solutions or no solution at all, depending on the specific system of linear equations.
- Option D: Matrix P does not need to be square for a solution to exist. The dimensions of matrix P, vector x, and vector q are independent of each other.
In conclusion, the correct answer is option A. For a solution to exist for the unknown vector x in the matrix equation Px = q, the augmented matrix [Pq] must have the same rank as matrix P.
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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)Augmented matrix [Pq] must have the same rank as matrix Pb)Vector q must have only non-zero elementsc)Matrix P must be singulard)Matrix P must be squareCorrect answer is option 'A'. Can you explain this answer?
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