SOLUTION: Hi all, I am struggling a bit with the following matrices problem. I need to find the determinate of the following matrices by Inspection. Then give reasons in each each. How do I

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Question 211750: Hi all, I am struggling a bit with the following matrices problem.
I need to find the determinate of the following matrices by Inspection. Then give reasons in each each. How do I do this?
The matrices are:
B =
( 2 11 -3 1)
( 1 5 -7 -4)
( 6 13 -5 2)
( 4 22 -6 2)
C =
( 2 9 -1)
( 0 2 7)
( 0 0 4)
D =
( -3 0 0)
( 0 2 0)
( 0 0 -5)
Any help would be great.
-Nick.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
For matrix B, the key is to notice that row 4 is exactly twice row 1. In such a matrix the determinant is always zero.

Both matrices C and D are diagonal matrices (i.e. a matrix with nothing but zeros above and/or below the main diagonal). In diagonal matrices the determinant is simply the product of the numbers in the diagonal. For C this would be 2%2A2%2A4+=+16. For D this would be -3%2A2%2A-5+=+30.

If you need more "reasons"...

For matrix B, think about adding -2 times the first row to the fourth row. This type of operation does not change the value of the determinant and, in this casse, it results in row 4 becoming all zeros. Then think about finding the value of the determinant using the fourth row. Result: 0.

For matrices C and D, think about using the first column to find the value of the determinant. Since there is only one non-zero value in this column, it should be easy to visualize the calculations and see why only the product of the numbers in the diagonal will matter.