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If M is a square matrix with a zero determinant, which of the following assertion(S) is (are) correct?
S1: Each row of M can be represented as a linear combination of the other rows.
S2: Each column of M can be represented as a linear combination of the other columns.
S3: MX = O has a nontrivial solution.
S4: M has an inverse.
S1: Each row of M can be represented as a linear combination of the other rows.
S2: Each column of M can be represented as a linear combination of the other columns.
S3: MX = O has a nontrivial solution.
S4: M has an inverse.
- a)S3 and S2 only
- b)S1 and S4 only
- c)S1 and S3 only
- d)S1, S2 and S3 only
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
If M is a square matrix with a zero determinant, which of the followin...
S1 and S2:
Since M has zero determinant, its rank is not full i.e. if M is of size 3 x 3 , then its rank is not 3. So there is a linear combination of rows which evaluates to 0 i.e. k1R1 + k2R2 +...+knRn = 0 and there is a linear combination of columns which evaluates to 0 i.e.
Now any row Ri can be written as linear combination of other rows as:
Similar is the case for columns.
So S1 and S2 are true.
Since M has zero determinant, its rank is not full i.e. if M is of size 3 x 3 , then its rank is not 3. So there is a linear combination of rows which evaluates to 0 i.e. k1R1 + k2R2 +...+knRn = 0 and there is a linear combination of columns which evaluates to 0 i.e.
Now any row Ri can be written as linear combination of other rows as:
Similar is the case for columns.
So S1 and S2 are true.
Most Upvoted Answer
If M is a square matrix with a zero determinant, which of the followin...
Explanation:
To determine which assertions are correct, we need to understand the properties of a square matrix with a zero determinant.
Property 1: A square matrix with a zero determinant is singular, which means it does not have an inverse.
Since an inverse does not exist for a matrix with zero determinant, assertion S4 is incorrect.
Property 2: A square matrix with a zero determinant means that its rows (or columns) are linearly dependent.
This means that at least one row (or column) can be represented as a linear combination of the other rows (or columns).
Therefore, assertions S1 and S2 are correct.
Property 3: A square matrix with a zero determinant implies that the homogeneous equation MX = O has a non-trivial solution, where X is a non-zero vector.
This means that there exists a non-zero vector X such that when multiplied by the matrix M, the result is the zero vector.
Therefore, assertion S3 is correct.
Summary:
Based on the properties of a square matrix with a zero determinant, the correct assertions are:
- S1: Each row of M can be represented as a linear combination of the other rows.
- S2: Each column of M can be represented as a linear combination of the other columns.
- S3: MX = O has a non-trivial solution.
The correct answer is option 'D': S1, S2, and S3 only.
To determine which assertions are correct, we need to understand the properties of a square matrix with a zero determinant.
Property 1: A square matrix with a zero determinant is singular, which means it does not have an inverse.
Since an inverse does not exist for a matrix with zero determinant, assertion S4 is incorrect.
Property 2: A square matrix with a zero determinant means that its rows (or columns) are linearly dependent.
This means that at least one row (or column) can be represented as a linear combination of the other rows (or columns).
Therefore, assertions S1 and S2 are correct.
Property 3: A square matrix with a zero determinant implies that the homogeneous equation MX = O has a non-trivial solution, where X is a non-zero vector.
This means that there exists a non-zero vector X such that when multiplied by the matrix M, the result is the zero vector.
Therefore, assertion S3 is correct.
Summary:
Based on the properties of a square matrix with a zero determinant, the correct assertions are:
- S1: Each row of M can be represented as a linear combination of the other rows.
- S2: Each column of M can be represented as a linear combination of the other columns.
- S3: MX = O has a non-trivial solution.
The correct answer is option 'D': S1, S2, and S3 only.
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If M is a square matrix with a zero determinant, which of the following assertion(S) is (are) correct?S1: Each row of M can be represented as a linearcombination of the other rows.S2: Each column of M can be represented as a linear combination of the other columns.S3: MX = O has a nontrivial solution.S4: M has an inverse.a)S3 and S2 onlyb)S1 and S4 onlyc)S1 and S3 onlyd)S1, S2 and S3 onlyCorrect answer is option 'D'. Can you explain this answer?
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If M is a square matrix with a zero determinant, which of the following assertion(S) is (are) correct?S1: Each row of M can be represented as a linearcombination of the other rows.S2: Each column of M can be represented as a linear combination of the other columns.S3: MX = O has a nontrivial solution.S4: M has an inverse.a)S3 and S2 onlyb)S1 and S4 onlyc)S1 and S3 onlyd)S1, S2 and S3 onlyCorrect answer is option 'D'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about If M is a square matrix with a zero determinant, which of the following assertion(S) is (are) correct?S1: Each row of M can be represented as a linearcombination of the other rows.S2: Each column of M can be represented as a linear combination of the other columns.S3: MX = O has a nontrivial solution.S4: M has an inverse.a)S3 and S2 onlyb)S1 and S4 onlyc)S1 and S3 onlyd)S1, S2 and S3 onlyCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If M is a square matrix with a zero determinant, which of the following assertion(S) is (are) correct?S1: Each row of M can be represented as a linearcombination of the other rows.S2: Each column of M can be represented as a linear combination of the other columns.S3: MX = O has a nontrivial solution.S4: M has an inverse.a)S3 and S2 onlyb)S1 and S4 onlyc)S1 and S3 onlyd)S1, S2 and S3 onlyCorrect answer is option 'D'. Can you explain this answer?.
If M is a square matrix with a zero determinant, which of the following assertion(S) is (are) correct?S1: Each row of M can be represented as a linearcombination of the other rows.S2: Each column of M can be represented as a linear combination of the other columns.S3: MX = O has a nontrivial solution.S4: M has an inverse.a)S3 and S2 onlyb)S1 and S4 onlyc)S1 and S3 onlyd)S1, S2 and S3 onlyCorrect answer is option 'D'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about If M is a square matrix with a zero determinant, which of the following assertion(S) is (are) correct?S1: Each row of M can be represented as a linearcombination of the other rows.S2: Each column of M can be represented as a linear combination of the other columns.S3: MX = O has a nontrivial solution.S4: M has an inverse.a)S3 and S2 onlyb)S1 and S4 onlyc)S1 and S3 onlyd)S1, S2 and S3 onlyCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If M is a square matrix with a zero determinant, which of the following assertion(S) is (are) correct?S1: Each row of M can be represented as a linearcombination of the other rows.S2: Each column of M can be represented as a linear combination of the other columns.S3: MX = O has a nontrivial solution.S4: M has an inverse.a)S3 and S2 onlyb)S1 and S4 onlyc)S1 and S3 onlyd)S1, S2 and S3 onlyCorrect answer is option 'D'. Can you explain this answer?.
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ample number of questions to practice If M is a square matrix with a zero determinant, which of the following assertion(S) is (are) correct?S1: Each row of M can be represented as a linearcombination of the other rows.S2: Each column of M can be represented as a linear combination of the other columns.S3: MX = O has a nontrivial solution.S4: M has an inverse.a)S3 and S2 onlyb)S1 and S4 onlyc)S1 and S3 onlyd)S1, S2 and S3 onlyCorrect answer is option 'D'. Can you explain this answer? tests, examples and also practice Computer Science Engineering (CSE) tests.
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