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Linear Algebra

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Co-Factor Expansion for Determinants

LINALG-OIKZHE

Suppose that $A$ is an $n\times n$ matrix and that the entry in the $i$th row and $j$th column of $A$ is given by $a_{ij}$. To compute the determinant of $A$ using a cofactor expansion across the $i$th row we calculate:

$$\det(A)=a_{i1} C_{i1}+a_{i2} C_{i2}+\ldots +a_{in}C_{in}$$

What is $C_{ij}$?

A

$C_{ij}$ is the determinant of the matrix $A_{ij}$, which is the matrix that results from deleting the $i$th row and the $j$th column of $A$.

B

$C_{ij}$ is product of $(-1)^{ij}$ and the determinant of the matrix $A_{ij}$, which is the matrix that results from deleting the $i$th row and the $j$th column of $A$.

C

$C_{ij}$ is product of $(-1)^{i+j}$ and the determinant of the matrix $A_{ij}$, which is the matrix that results from deleting the $i$th row and the $j$th column of $A$.

D

$C_{ij}$ is product of $(-1)^{i+j}$ and the determinant of the matrix $A_{ij}$ which is the matrix that results from deleting the $j$th row and the $i$th column of $A$.

E

None of the above.