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Linear Algebra

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Moderate

$n$-linearity of Determinant

LINALG-JZGV2E

If:

$$A=\begin{pmatrix} a_{11} & a_{12} & a_{13} \\\ a_{21} & a_{22} & a_{23} \\\ a_{31} &a_{32} & a_{33}\end{pmatrix}$$

...and:

$$A'=\begin{pmatrix} a_{11} & b_{12} & a_{13} \\\ a_{21} & b_{22} & a_{23} \\\ a_{31} &b_{32} & a_{33}\end{pmatrix}$$

...and:

$$A''=\begin{pmatrix} a_{11} & a_{12}+b_{12} & a_{13} \\\ a_{21} & a_{22}+b_{22} & a_{23} \\\ a_{31} &a_{32}+b_{32} & a_{33}\end{pmatrix}$$

...where $det(A)=3$ and $det(A')=4$ then $det(A'')=...$

A

$12$

B

$7$

C

$-1$

D

$1$