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.