It is given that determinant of the below matrix is -3\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}
What is the determinant of the following:
\begin{bmatrix}-8d+a&-8e+b&-8f+c\\d&e&f\\g&h&i\end{bmatrix}
I know that swapping rows negates the determinant, and multiplying a row by a scalar scales the determinant. But I can't get this question correct. I thought it would be 24, because adding one row to another shouldn't affect the determinant, only the multiplication by -8 would, so the determinant would be -8 * -3 = 24. But that isn't the correct answer, and neither is -24. Can someone help me understand what I'm missing?
Best Answer
Although the problem has been solved for the OP apparently, I would add a more concise answer for future readers:
Invariance Property: The determinant is unchanged if any row (or column) is multiplied by a scalar and then added or subtracted from another row (or column).
Basically this translates to
$$\text{det}\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=\text{det}\begin{bmatrix}a+kd&b+kd&c+kd\\d&e&f\\g&h&i\end{bmatrix}$$
Why is this true?
Can you follow it up from here?