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For a square matrix A = [aij] the quantity calculated for any element aij in A as the determinant of thesquare sub-matrix of order (n – 1) obtained by leaving the ith row and jth column of A is known as:
- a)Cofactor
- b)Minor
- c)Coefficient
- d)Elements
Correct answer is option 'B'. Can you explain this answer?
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For a square matrix A = [aij] the quantity calculated for any element ...
Understanding Minors in Matrices
In the context of a square matrix A = [aij], the term "minor" is crucial for various calculations in linear algebra, particularly in finding determinants and inverses.
What is a Minor?
- A minor of an element aij in matrix A is defined as the determinant of the square sub-matrix formed by deleting the ith row and jth column from A.
- This sub-matrix has an order of (n - 1), where n is the order of the original matrix A.
Significance of Minors
- Minors play a vital role in calculating the determinant of larger matrices.
- The concept of minors is foundational in the computation of cofactors and adjugates, which are used in finding the matrix inverse.
Example for Clarity
- Consider a 3x3 matrix A:
| a11 a12 a13 |
| a21 a22 a23 |
| a31 a32 a33 |
- To find the minor M11 of the element a11, you would take the determinant of the matrix formed by removing the first row and the first column:
| a22 a23 |
| a32 a33 |
- The determinant of this 2x2 matrix is the minor M11.
Conclusion
Thus, the correct answer to the question is option 'B' - Minor, as it refers specifically to the determinant of the sub-matrix formed by removing the respective row and column associated with an element in the matrix. Understanding minors is essential for mastering the concepts of determinants and matrix theory in linear algebra.
In the context of a square matrix A = [aij], the term "minor" is crucial for various calculations in linear algebra, particularly in finding determinants and inverses.
What is a Minor?
- A minor of an element aij in matrix A is defined as the determinant of the square sub-matrix formed by deleting the ith row and jth column from A.
- This sub-matrix has an order of (n - 1), where n is the order of the original matrix A.
Significance of Minors
- Minors play a vital role in calculating the determinant of larger matrices.
- The concept of minors is foundational in the computation of cofactors and adjugates, which are used in finding the matrix inverse.
Example for Clarity
- Consider a 3x3 matrix A:
| a11 a12 a13 |
| a21 a22 a23 |
| a31 a32 a33 |
- To find the minor M11 of the element a11, you would take the determinant of the matrix formed by removing the first row and the first column:
| a22 a23 |
| a32 a33 |
- The determinant of this 2x2 matrix is the minor M11.
Conclusion
Thus, the correct answer to the question is option 'B' - Minor, as it refers specifically to the determinant of the sub-matrix formed by removing the respective row and column associated with an element in the matrix. Understanding minors is essential for mastering the concepts of determinants and matrix theory in linear algebra.
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For a square matrix A = [aij] the quantity calculated for any element ...
The minor of an element aij in A is calculated as the determinant of the square sub-matrix of order (n-1) obtained by leaving the ith row and jth column of A.
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For a square matrix A = [aij] the quantity calculated for any element aij in A as the determinant of thesquare sub-matrix of order (n – 1) obtained by leaving the ith row and jth column of A is known as:a)Cofactorb)Minorc)Coefficientd)ElementsCorrect answer is option 'B'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about For a square matrix A = [aij] the quantity calculated for any element aij in A as the determinant of thesquare sub-matrix of order (n – 1) obtained by leaving the ith row and jth column of A is known as:a)Cofactorb)Minorc)Coefficientd)ElementsCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a square matrix A = [aij] the quantity calculated for any element aij in A as the determinant of thesquare sub-matrix of order (n – 1) obtained by leaving the ith row and jth column of A is known as:a)Cofactorb)Minorc)Coefficientd)ElementsCorrect answer is option 'B'. Can you explain this answer?.
For a square matrix A = [aij] the quantity calculated for any element aij in A as the determinant of thesquare sub-matrix of order (n – 1) obtained by leaving the ith row and jth column of A is known as:a)Cofactorb)Minorc)Coefficientd)ElementsCorrect answer is option 'B'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about For a square matrix A = [aij] the quantity calculated for any element aij in A as the determinant of thesquare sub-matrix of order (n – 1) obtained by leaving the ith row and jth column of A is known as:a)Cofactorb)Minorc)Coefficientd)ElementsCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a square matrix A = [aij] the quantity calculated for any element aij in A as the determinant of thesquare sub-matrix of order (n – 1) obtained by leaving the ith row and jth column of A is known as:a)Cofactorb)Minorc)Coefficientd)ElementsCorrect answer is option 'B'. Can you explain this answer?.
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