Can Someone help me with these two algebra problems?

1. We can take out common factors: Take out 4 from the first column, 2 from the 2nd columns and 3 from the last column. You get 4*2*3 = 24 and the determinant left to find is:

1 2 -2

1 -2 1

1 1 2

Next you can do row and column operations without changing the value of the determinant so, subtract row 1 from row 2 and from row 3 to get:

1 2 -2

0 -4 3

0 -1 4

Now evaluate along the first column to get:

1*((-4)*4 - (-1)*3) - 0(2*4 - (-1)*(-2)) + 0*(2*3 - (-2)*(-4)) = -13 - 0 + 0 = -

-13

So the value of the determinant is 24*(-13) = -312 which is answer C.

Remember that with determinants we can:

1. Take out common factors from rows and columns.

2. Do row and column operations without changing the value of the determinant.

3. Swap rows and columns, which will change the sign of the determinant.

The best way to solve them is to try to reduce the determinant to have a column or row with as many 0 as we can, so we don't have to do much work when opening the determinant up.

2. We can construct a system of equations:

say that we use A from alloy one alloy and B from the other alloy we get the following system of equations:

0.1A + 0.4B = 65 (this is for gold)

0.2A + 0.5B = 118 (this is for lead)

We solve to get that:

0.5B - 0.8B = 118 - 130

Therefore B = 40g

We can now find A:

0.2A = 118 - 0.5B = 118 - 20 = 98

Hence A = 490g

So the answer is C.

Hope it helped :-)