# If a 4x4 matrix `A` with rows` v_1,v_2,v_3` and `v_4` has determinant `det(A) = -8` , then `det[[2v_1+8v_3],[v_2],[9v_1+3v_3],[v_4]] =?`

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### 1 Answer

We will use the following two properties (axioms) of determinant:

1) Multiplying a row by `k` multiplies the determinant by `k.`

`det(v_1,ldots,k cdot v_i,ldots,v_n)=k cdot det(v_1,ldots,v_i,ldots,v_n)`

2) Adding one row multiplied by `k` to another row does not change the determinant.

`det(v_1,ldots,v_i+k cdot v_j,ldots,v_n)=det(v_1,ldots,v_i,ldots,v_n)`

Let's first apply second property.

`det(2v_1+8v_3,v_2,9v_1+3v_3,v_4)=det(2v_1,v_2,3v_3,v_4)`

Now we use the first property.

`=2det(v_1,v_2,3v_3,v_4)=3cdot2cdot det(v_1,v_2,v_3,v_4)=`

`6(-8)=-48` **<--- Your solution**