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Questions and Answers
What are the elementary transformations of a matrix?
What are the elementary transformations of a matrix?
How many types of elementary transformations are due to rows in a matrix?
How many types of elementary transformations are due to rows in a matrix?
What does R1 ↔ R2 mean in the context of a matrix?
What does R1 ↔ R2 mean in the context of a matrix?
What operation is denoted by C1 ↔ C2 in matrix transformations?
What operation is denoted by C1 ↔ C2 in matrix transformations?
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In the given matrix, if A = [[1, 2], [3, 4]], what will be the result after the operation R1 ↔ R2?
In the given matrix, if A = [[1, 2], [3, 4]], what will be the result after the operation R1 ↔ R2?
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Which branch of science extensively uses matrices for problem solving?
Which branch of science extensively uses matrices for problem solving?
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What is the language of atomic Physics, according to the text?
What is the language of atomic Physics, according to the text?
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Where are matrices NOT commonly used according to the text?
Where are matrices NOT commonly used according to the text?
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'Recall that R and C symbolically represent the rows and columns of a matrix.' What are R and C symbols for in matrix operations?
'Recall that R and C symbolically represent the rows and columns of a matrix.' What are R and C symbols for in matrix operations?
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'It is necessary to learn the uses of matrices with the help of elementary transformations and the inverse of a matrix.' What is highlighted as essential for learning about matrices?
'It is necessary to learn the uses of matrices with the help of elementary transformations and the inverse of a matrix.' What is highlighted as essential for learning about matrices?
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Study Notes
Determinants and Minors
- Minor of an element (a_{ij}) is obtained by deleting the (i)-th row and (j)-th column of a determinant.
- Denoted as (M_{ij}).
- For a determinant of order (n), the order of the minor is (n-1).
Co-factors
- Co-factor of an element (a_{ij}) is computed as (A_{ij} = (-1)^{i+j} M_{ij}).
- This means the sign of the co-factor depends on the sum of the indices (i) and (j).
Adjoint of a Matrix
- Adjoint of a matrix (A) is the transpose of the co-factor matrix ([A_{ij}]).
- Denoted as (\text{adj } A).
Example Analysis
- For a matrix (A): [ A = \begin{bmatrix} 1 & -2 & 3 \ 4 & 5 & -6 \ 7 & -8 & 9 \end{bmatrix} ]
- Determinant (|A|) can be calculated from elements.
- To find the minor (M_{21}) (element 4), delete the 2nd row and 1st column: [ M_{21} = \begin{vmatrix} -2 & 3 \ -8 & 9 \end{vmatrix} = -2 \times 9 - (-8 \times 3) = 6 ]
Co-factor Calculation Example
- Co-factor (A_{21}) for element (4): [ A_{21} = (-1)^{2+1} \cdot M_{21} = -6 ]
Steps to Find Adjoint
- For a square (3 \times 3) matrix, calculate the co-factors for each element to form the co-factor matrix.
- For example: [ A = \begin{bmatrix} 2 & -3 \ 4 & 1 \end{bmatrix} ]
- Co-factors calculated:
- (A_{11} = 1)
- (A_{12} = -4)
- (A_{21} = 3)
- (A_{22} = 2)
- The co-factor matrix becomes: [ \begin{bmatrix} 1 & -4 \ 3 & 2 \end{bmatrix} \rightarrow \text{adj } A = \begin{bmatrix} 1 & 3 \ -4 & 2 \end{bmatrix} ]
Additional Example
- For matrix (A): [ A = \begin{bmatrix} 2 & 0 & -1 \ 3 & 1 & 2 \ -1 & 1 & 2 \end{bmatrix} ]
- Various elements lead to respective minors and co-factors.
- Finally, arranged into co-factor matrix leading to adjoint: [ \text{adj } A = \begin{bmatrix} 0 & -1 & 1 \ -8 & 3 & -7 \ 4 & -2 & 2 \end{bmatrix} ]
Determinant Expansion
- A determinant can be expanded using any row or column, allowing flexibility in computation.
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Description
Learn about minors and co-factors of elements in determinants, including how to calculate them and their significance in matrix operations. Explore the relationship between minors, co-factors, and the adjoint of a matrix.
