Minors and Cofactors are important to calculate the adjoint and inverse of a matrix. As the name suggests, a Minor is a smaller part of the larger matrix obtained for a particular element of the matrix by deleting the terms of the row and column to which the element belongs. Cofactor is (-1)i+j times the minor of the matrix. They are the backbones of Linear Algebra and are used to find the value of the determinant, adjoint, and inverse of a matrix
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Answer:
Sure let's go into more detail about minors and cofactors.
A minor of a square matrix A is the determinant of a smaller matrix formed by deleting one or more rows and columns from A. To find the minor of an element at position (i j) in matrix A we remove the i-th row and j-th column and then calculate the determinant of the resulting matrix.
For example let's say we have a 3x3 matrix A:
A =
| a11 a12 a13 |
| a21 a22 a23 |
| a31 a32 a33 |
To find the minor of a22 we remove the second row and second column:
minor(a22) =
| a11 a13 |
| a31 a33 |
The cofactor of an element at position (i j) in matrix A is defined as (-1)^(i+j) times the minor of A at position (i j). It is denoted as C(i j).
For example let's find the cofactor of a22 using the minor we calculated earlier:
cofactor(a22) = (-1)^(2+2) * minor(a22) = (-1)^4 * minor(a22) = 1 * minor(a22) = minor(a22)
So in this case the cofactor of a22 is equal to the minor of a22.
The cofactors of all elements in a matrix can be arranged in a matrix called the cofactor matrix or the matrix of cofactors. In this matrix the element at position (i j) is the cofactor C(i j) of the element at position (i j) in the original matrix.
For example let's find the cofactor matrix of matrix A:
Cofactor matrix of A =
| C(11) C(12) C(13) |
| C(21) C(22) C(23) |
| C(31) C(32) C(33) |
Once we have the cofactor matrix we can use it to calculate the adjoint and inverse of a matrix.
The adjoint of a matrix A is the transpose of its cofactor matrix. It is denoted as adj(A) or A*.
The inverse of a square matrix A is given by the formula:
inverse(A) = adj(A) / det(A)
Where det(A) is the determinant of A.
Overall minors and cofactors play a crucial role in finding the determinant adjoint and inverse of a matrix providing us with valuable information about the properties of the matrix.
Answer:
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Step-by-step explanation:
Practice Problems on How to Find Minors and Cofactors
Question 1: Find the cofactor of a12 in the following.
Solution:
In this problem, we have to find the cofactor of a12, therefore, eliminate all the elements of the first row and the second column and by obtaining the determinant of the remaining elements, we can calculate the cofactor of a12
Here, a12= Element of the first row and second column = –3
M12 = Minor of a12
= 6 (-7) – 4(1)
= -42 – 4 = -46
Cofactor of (-3) = (-1)1+2 (-46) = -(-46) = 46