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Let A be a 3 × 3 matrix with rank 2. Then AX = 0 has
- a)Only the trivial solution X = 0
- b)One independent solution
- c)Two independent solutions
- d)Three independent solutions
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Let A be a 3 × 3 matrix with rank 2. Then AX = 0 hasa)Only the tr...
(b)
We know , rank (A) + Solution space X(A) = no. of unknowns.
⇒2 + X(A) = 3 . [Solution space X(A)= No. of linearly independent vectors]
⇒ X(A) =1.
We know , rank (A) + Solution space X(A) = no. of unknowns.
⇒2 + X(A) = 3 . [Solution space X(A)= No. of linearly independent vectors]
⇒ X(A) =1.
Most Upvoted Answer
Let A be a 3 × 3 matrix with rank 2. Then AX = 0 hasa)Only the tr...
Explanation:
To understand why the correct answer is option 'B', let's break down the problem and analyze it step by step.
Step 1: Understanding Rank
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. In other words, it represents the dimension of the vector space spanned by the rows or columns of the matrix.
In this case, we are given that matrix A has a rank of 2. This means that there are two linearly independent rows or columns in matrix A.
Step 2: Solving AX = 0
The equation AX = 0 represents a homogeneous system of linear equations, where X is a vector of variables and 0 is the zero vector.
Since A is a 3x3 matrix, X is a vector with three variables. Therefore, the equation AX = 0 represents a system of three linear equations with three variables.
Step 3: The Rank-Nullity Theorem
The Rank-Nullity Theorem states that for any matrix A, the sum of the rank and nullity of A is equal to the number of columns in A.
In this case, since A is a 3x3 matrix, the number of columns is 3. And we are given that the rank of A is 2. Therefore, the nullity of A (the number of independent solutions to AX = 0) can be calculated as:
Nullity = Number of columns - Rank
Nullity = 3 - 2
Nullity = 1
Step 4: Conclusion
Based on the Rank-Nullity Theorem, we have determined that the nullity of A is 1. This means that the equation AX = 0 has one independent solution.
Therefore, the correct answer is option 'B' - One independent solution.
To understand why the correct answer is option 'B', let's break down the problem and analyze it step by step.
Step 1: Understanding Rank
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. In other words, it represents the dimension of the vector space spanned by the rows or columns of the matrix.
In this case, we are given that matrix A has a rank of 2. This means that there are two linearly independent rows or columns in matrix A.
Step 2: Solving AX = 0
The equation AX = 0 represents a homogeneous system of linear equations, where X is a vector of variables and 0 is the zero vector.
Since A is a 3x3 matrix, X is a vector with three variables. Therefore, the equation AX = 0 represents a system of three linear equations with three variables.
Step 3: The Rank-Nullity Theorem
The Rank-Nullity Theorem states that for any matrix A, the sum of the rank and nullity of A is equal to the number of columns in A.
In this case, since A is a 3x3 matrix, the number of columns is 3. And we are given that the rank of A is 2. Therefore, the nullity of A (the number of independent solutions to AX = 0) can be calculated as:
Nullity = Number of columns - Rank
Nullity = 3 - 2
Nullity = 1
Step 4: Conclusion
Based on the Rank-Nullity Theorem, we have determined that the nullity of A is 1. This means that the equation AX = 0 has one independent solution.
Therefore, the correct answer is option 'B' - One independent solution.
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Let A be a 3 × 3 matrix with rank 2. Then AX = 0 hasa)Only the trivial solution X = 0b)One independent solutionc)Two independent solutionsd)Three independent solutionsCorrect answer is option 'B'. Can you explain this answer?
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