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A determinant is unaltered , if
- a)Every element in a column is multiplied by the same factor
- b)Two rows are interchanged
- c)Two columns are interchanged
- d)To each element of any row is added the corresponding element of the other row multiplied by a given factor
Correct answer is option 'D'. Can you explain this answer?
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A determinant is unaltered , ifa)Every element in a column is multipli...
This is because of the elementary transformations of determinants . The value of determinant remains unaffected by applying elementary transformations.
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A determinant is unaltered , ifa)Every element in a column is multipli...
The Effect of Operations on Determinants:
Determinants are mathematical objects that are used to solve systems of linear equations and to calculate the areas and volumes of geometric shapes. They have various properties that help in manipulating and solving equations. One important property is that a determinant remains unaltered under certain operations. Let's discuss these operations in detail:
Multiplying Every Element in a Column by the Same Factor:
If we multiply every element in a column of a determinant by the same factor, the value of the determinant remains the same. This is because when we expand the determinant, each term will be multiplied by the same factor, and these factors will cancel out when we calculate the determinant. Therefore, option A is incorrect.
Interchanging Two Rows:
If we interchange two rows of a determinant, the sign of the determinant changes. This means that if the original determinant is positive, the new determinant will be negative, and vice versa. Therefore, option B is incorrect.
Interchanging Two Columns:
If we interchange two columns of a determinant, the sign of the determinant also changes. Therefore, option C is incorrect.
Adding a Multiple of One Row to Another Row:
This operation is called row operation. If we add a multiple of one row to another row, the determinant remains unaltered. This is because each term in the determinant will have two elements from the same row and one element from the other row. When we perform the row operation, the term with the multiple will cancel out, leaving the determinant unchanged. Therefore, option D is the correct answer.
In conclusion, the determinant of a matrix is unaltered if we add a multiple of one row to another row. This property is useful in solving systems of linear equations and performing various calculations involving determinants.
Determinants are mathematical objects that are used to solve systems of linear equations and to calculate the areas and volumes of geometric shapes. They have various properties that help in manipulating and solving equations. One important property is that a determinant remains unaltered under certain operations. Let's discuss these operations in detail:
Multiplying Every Element in a Column by the Same Factor:
If we multiply every element in a column of a determinant by the same factor, the value of the determinant remains the same. This is because when we expand the determinant, each term will be multiplied by the same factor, and these factors will cancel out when we calculate the determinant. Therefore, option A is incorrect.
Interchanging Two Rows:
If we interchange two rows of a determinant, the sign of the determinant changes. This means that if the original determinant is positive, the new determinant will be negative, and vice versa. Therefore, option B is incorrect.
Interchanging Two Columns:
If we interchange two columns of a determinant, the sign of the determinant also changes. Therefore, option C is incorrect.
Adding a Multiple of One Row to Another Row:
This operation is called row operation. If we add a multiple of one row to another row, the determinant remains unaltered. This is because each term in the determinant will have two elements from the same row and one element from the other row. When we perform the row operation, the term with the multiple will cancel out, leaving the determinant unchanged. Therefore, option D is the correct answer.
In conclusion, the determinant of a matrix is unaltered if we add a multiple of one row to another row. This property is useful in solving systems of linear equations and performing various calculations involving determinants.
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A determinant is unaltered , ifa)Every element in a column is multiplied by the same factorb)Two rows are interchangedc)Two columns are interchangedd)To each element of any row is added the corresponding element of the other row multiplied by a given factorCorrect answer is option 'D'. Can you explain this answer?
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A determinant is unaltered , ifa)Every element in a column is multiplied by the same factorb)Two rows are interchangedc)Two columns are interchangedd)To each element of any row is added the corresponding element of the other row multiplied by a given factorCorrect answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A determinant is unaltered , ifa)Every element in a column is multiplied by the same factorb)Two rows are interchangedc)Two columns are interchangedd)To each element of any row is added the corresponding element of the other row multiplied by a given factorCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A determinant is unaltered , ifa)Every element in a column is multiplied by the same factorb)Two rows are interchangedc)Two columns are interchangedd)To each element of any row is added the corresponding element of the other row multiplied by a given factorCorrect answer is option 'D'. Can you explain this answer?.
A determinant is unaltered , ifa)Every element in a column is multiplied by the same factorb)Two rows are interchangedc)Two columns are interchangedd)To each element of any row is added the corresponding element of the other row multiplied by a given factorCorrect answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A determinant is unaltered , ifa)Every element in a column is multiplied by the same factorb)Two rows are interchangedc)Two columns are interchangedd)To each element of any row is added the corresponding element of the other row multiplied by a given factorCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A determinant is unaltered , ifa)Every element in a column is multiplied by the same factorb)Two rows are interchangedc)Two columns are interchangedd)To each element of any row is added the corresponding element of the other row multiplied by a given factorCorrect answer is option 'D'. Can you explain this answer?.
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