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Definitions
Instructions: Study and familiarize
yourself with the following definitions.
A determinant is a number associated with a square matrix.
Cramer's Rule is a formula method for solving a system of linear equations.
The determinant of any two-by-two matrix A
is defined as a number det(A)=ad - bc
The determinant of order n
is the determinant of a matrix n-by-n.
The minor of an element in a determinant
is the (smaller) determinant that remains after deleting all entries in its row and column.
Example
|
The minor of element a21 is the determinant
of the matrix
|
The minor of element a32 is determinant
of the matrix
,b
|
The cofactor
of an entry aij is (-1)i+j
multiplied by the minor of the aij entry.
This says that if the sum of the row and
column is even, the cofactor is the same as the minor.
If the
sum of the row and column is odd, the cofactor of that
entry is the opposite of its minor.
Determinant Expansion Theorem
A determinant of order n
is a real number whose value is the
sum of the products obtained by multiplying each element
of a row (or column) by its cofactor.
Example:
The determinant expanded about the second row of the matrix A
is equal to
det(A) =
(1)(-1)det(A21) + (+2)(+1)det(A22) +
(3)(-1)det(A23)= -8 +2(4) + 3(-1) = 3
where A21, A22,
A23 are matrices
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