- #1
ajayguhan
- 153
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In a matrix the sum of element of a matrix in a row times it's co factor of that elemt gives the determinant value, but why does the sum of element of a matrix times cofactor of different row is always zero?
A matrix is a mathematical structure that consists of rows and columns of numbers. It is commonly used in linear algebra and has many applications in fields such as physics, engineering, and computer graphics.
A cofactor is a number that is calculated from the elements of a matrix. It is used in various matrix operations, such as finding the inverse of a matrix or solving systems of linear equations.
This is because of the properties of determinants, which are closely related to cofactors. When we expand a determinant along a particular row or column, we can see that the elements are multiplied by their cofactors. Since the determinant of a matrix with all zeros in a particular row or column is zero, the sum of the elements multiplied by their cofactors must also be zero.
This property is important because it allows us to solve systems of linear equations using the inverse of a matrix. By finding the cofactors of a matrix and using them to form the adjugate matrix, we can find the inverse of the original matrix. This, in turn, allows us to solve systems of linear equations, which have many real-world applications.
Yes, this property holds true for matrices of any size. The sum of matrix elements multiplied by their cofactors will always equal zero, as long as the matrix is square (has the same number of rows and columns).