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Questions and Answers
What are the elementary transformations of a matrix?
What are the elementary transformations of a matrix?
- Interchange of rows or columns (correct)
- Negation and inversion
- Multiplication and division
- Addition and subtraction
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?
- Three (correct)
- One
- Two
- Four
What does R1 ↔ R2 mean in the context of a matrix?
What does R1 ↔ R2 mean in the context of a matrix?
- Multiplication of rows 1 and 2
- Subtraction of row 1 from row 2
- Addition of rows 1 and 2
- Interchange of rows 1 and 2 (correct)
What operation is denoted by C1 ↔ C2 in matrix transformations?
What operation is denoted by C1 ↔ C2 in matrix transformations?
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?
Which branch of science extensively uses matrices for problem solving?
Which branch of science extensively uses matrices for problem solving?
What is the language of atomic Physics, according to the text?
What is the language of atomic Physics, according to the text?
Where are matrices NOT commonly used according to the text?
Where are matrices NOT commonly used according to the text?
'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?
'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?
Flashcards
Minor of an element (M_{ij})
Minor of an element (M_{ij})
The minor of an element in a matrix is found by deleting the row and column containing that element. It's essentially a smaller determinant obtained from the original.
Co-factor of an element (A_{ij})
Co-factor of an element (A_{ij})
The co-factor of an element a_{ij} in a matrix is calculated by multiplying its minor (M_{ij}) by (-1)^(i+j). This sign depends on the position of the element.
Adjoint of a matrix (adj A)
Adjoint of a matrix (adj A)
The adjoint of a matrix A is obtained by transposing its co-factor matrix. The co-factor matrix is formed by replacing each element with its corresponding co-factor.
Order of a minor
Order of a minor
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Determinant expansion
Determinant expansion
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Calculating the determinant
Calculating the determinant
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Co-factor sign
Co-factor sign
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Importance of minors, co-factors, and adjoints
Importance of minors, co-factors, and adjoints
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Transposing the co-factor matrix
Transposing the co-factor matrix
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Adjoint and matrix inverse
Adjoint and matrix inverse
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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.
