If the determinant of a 5x5 matrix?

Question

If the determinant of a 5x5 matrix A is det(A)=6, and the matrix B is obtained from A by multiplying the second column by 9 then det(B) is...

I'm really confused on how to do these sorts of problems. Thanks for the help!!

Answer 1

If you multiply a column or a row of a matrix by a constant, the determinant is multiplied by the same constant. So in this case det(B) = 54.

Remember that the determinant is the sum of a whole bunch of products containing one element from each column and row. If you multiply a single row or column by c, every one of those products will have exactly one element from that row or column and so the product will also be multiplied by c. So the whole determinant will be multiplied by c.

It follows from this that if you multiply a whole n×n matrix by c, you multiply the determinant by c^n.

Answer 2

det(B)=54. Think of the way you'd caclulate a determinant by hand if the matrix was 3x3 or bigger:

You could pick the cell in the first row of second column. Let's say that the number is "a". Cross out everything else in the first row and second column, leaving you with a smaller matrix. Then you'd multiply the determinant of this by -a. Now go down the column and repeat the process. What you end up with is something like:
-a*(det of some 4x4 matrix) + b*(det of some 4x4 matrix) - c*(det of some 4x4 matrix) + d*(det of some 4x4 matrix) - e*(det of some 4x4 matrix).

where a, b, c, d and e are the numbers in the second column. If all of these were multiplied by 9, then
det(B) = -9a*(det of some 4x4 matrix) + 9b*(det of some 4x4 matrix) - 9c*(det of some 4x4 matrix) + 9d*(det of some 4x4 matrix) - 9e*(det of some 4x4 matrix). =
9 [ -a*(det of some 4x4 matrix) + b*(det of some 4x4 matrix) - c*(det of some 4x4 matrix) + d*(det of some 4x4 matrix) - e*(det of some 4x4 matrix) ] =
9 det(A) =
9*6 = 54

Answer 3

Why don't you do the same type of operation on a 2x2 or a 3x3 matrix?

It will yield the same general result.