Difference between revisions of "Minors and Cofactors"
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Delete the elements in <math>2</math><sup>nd</sup> row and <math>1</math><sup>st</sup> column , the remaining elements will be the minor of <math>4</math> represented by <math>M_{21}</math> | Delete the elements in <math>2</math><sup>nd</sup> row and <math>1</math><sup>st</sup> column , the remaining elements will be the minor of <math>4</math> represented by <math>M_{21}</math> | ||
<math>M_{21}=\begin{vmatrix} | <math>M_{21}=\begin{vmatrix} 2& 3 \\ 8&9 \end{vmatrix}= 2 \times 9 - 3 \times 8=18-24=-6</math> | ||
== Cofactors == | == Cofactors == | ||
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Cofactor of an element <math>a_{21}</math>, denoted by <math>A_{21}</math> is defined by <math>A_{21}=(-1)^{2+1}M_{21}</math> , where <math>M_{21}</math> is minor of <math>a_{21}</math> | Cofactor of an element <math>a_{21}</math>, denoted by <math>A_{21}</math> is defined by <math>A_{21}=(-1)^{2+1}M_{21}</math> , where <math>M_{21}</math> is minor of <math>a_{21}</math> | ||
We know from the above <math>M_{21}=- | We know from the above <math>M_{21}=-6</math> | ||
<math>A_{21}=(-1)^{2+1} \times - | <math>A_{21}=(-1)^{2+1} \times -6=(-1)^3 \times -6=-1 \times -6=6</math> |
Latest revision as of 07:44, 23 January 2024
Minors and cofactors are two of the most important concepts in matrices. They are essential in finding the adjoint and the inverse of a matrix.
Minors and cofactors are defined for each element of the matrix.
Minors
The minor of an element of the matrix is equal to the determinant of the remaining elements of the matrix, obtained after deleting the row and column containing the particular element in the matrix.
Minor of an element of a determinant is the determinant obtained by deleting its th row and th column in which element occurs. Minor of an element is denoted by
Example
Find the minor of element in the determinant
The element occurs in nd row and st column.
Delete the elements in nd row and st column , the remaining elements will be the minor of represented by
Cofactors
The cofactor of an element of the matrix is obtained by multiplying the minor of the element with to power of the sum of th and th column containing the element.
Cofactor of an element , denoted by is defined by = , where is minor of
Example
Find the cofactor of element in the determinant
The element occurs in nd row and st column. Hence
Cofactor of an element , denoted by is defined by , where is minor of
We know from the above