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Linear Algebra

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Moderate

Same Determinant or Not

LINALG-0YYUNY

Suppose that $A$ is a square, invertible matrix, with at least four rows. Which of the following must have the same determinant as $A$? Select ALL that apply.

A

The matrix $B$ formed by interchanging row one and row three of $A$.

B

The matrix $C$ formed by interchanging row one and row two, and then row three and row four of $A$.

C

The transpose of $A$.

D

The matrix $D$ formed by replacing row two with the sum of row two and row three.

E

$A^{-1}$

F

The matrix $E$ formed by replacing the first row of $A$ with a row of all zeros.

G

None of the above must have the same determinant as $A$.