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Row moves on a matrix

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A matrix $A$ of size $3 \times 3$ has $det (A) = 7$ . Matrix $B$ is built from $A$ in $3$ moves, in this order: swap rows $1$ and $3$ ; multiply the new row $2$ by $4$ ; add the new row $1$ , multiplied by $5$ , to the new row $3$ . What is $det (B)$ ?

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Hint

Each kind of row operation changes the determinant in its own fixed way; follow the moves one at a time.

Solution

Swapping $2$ rows flips the sign of the determinant, so $7$ becomes $- 7$ .
Multiplying one row by $4$ multiplies the determinant by $4$ , giving $- 7 \times 4 = - 28$ .
Adding a multiple of one row to another row leaves the determinant unchanged.
So $det (B) = - 28$ .

Answer: $- 28$

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