How would I answer the following questions on the determinants of matrices?

[sktsasus]

"1) If the determinant of a 5 x 5 matrix A is det(A) = 8, and the matrix B is obtained from A by multiplying the second row by 2, then det(B) = ?

2) If the determinant of a 4 x 4 matrix A is det(A) = 5, and the matrix C is obtained from A by swapping the third and fourth rows, then det(C) = ?

3) If the determinant of a 5 x 5 matrix A is det(A) = 6, and the matrix D is obtained from A by adding 4 times the third row to the second, then det(D) = ?"

My problem is not getting the new determinant but rather finding a matrix that satisfies the original determinant. Is there a formula through which I can find the matrix given its determinant, because computing the determinant of the new matrix should not be a problem.

Thank you!

 

[gregk]

My problem is not getting the new determinant but rather finding a matrix that satisfies the original determinant. Is there a formula through which I can find the matrix given its determinant, because computing the determinant of the new matrix should not be a problem.

Well, the question isn't asking you to actually find such a matrix is it? It just wants to know how the given row operation affects the determinant. You know the determinant of $\displaystyle A$, so the determinant of $\displaystyle B$ should be ... (use the relevant property of determinants and fill in the answer). Each of these questions can be answered without ever knowing what the matrix $\displaystyle A$ actually is.

 

[tkhunny]

Why does it have to be 5x5? Why not try the operations out on a 2x2 and see what happens. Of course, you'll have to alter the third one a bit.