CHAPTER 04.11: UNARY MATRIX OPERATIONS: Theorems on determinants Part 3 of 4
Let’s talk about some more theorems about determinants. Let A be a n by n matrix. If a column or row of A is multiplied by a scalar “k” – so what we’re doing here is that we’re taking a real number “k” and multiplying some column or some row by that number “k” and if a column or row of A is multiplied by a scalar “k” resulting in matrix B, then the determinant of B is that same as “k” times the determinant of A. So if you are able to recognize that – hey, another matrix is simply that one of the rows or columns has been multiplied by a scalar “k”, you don’t have to find the determinant of B by using the minors or other methods. You can simple say the determinant of B is simple “k” times the determinant of A. Let’s take an example.
Let’s suppose that somebody says: hey, I’m giving an A matrix which is 25, 5, 1, 64, 8, 1, 144, 12, 1. So somebody says this is the A matrix and somebody also then goes and gives you another B matrix and says that hey it will be 25, 6, 1, 64, 9.6, 1, 144, 14.4, 1. And you can see that the A matrix and B matrix have some similarities. The first column is the same as the first column right here. The third column is the same as a third column here, but what you’re finding out is that in the second column, the second column here is 1.2 times the second column right here. Because 1.2 times 5 is equal to 6, 1.2 times 8 is equal to 96, and 1.2 times 12 is 14.2. So based on that, we say that he determinant of B is 1.2 times the determinant of A. So we know what the determinant of A is then we can find out what the determinant of B is.
So for example, the determinant of this matrix is -84. So I can say that – hey, the determinant of B is 1.2 times -8.4. So I don’t have to find the determinant of this matrix by using the elementary methods to be able to do that. I could simple say the determinant of B is simple 1.2 times the determinant of A. And that’s the end of the segment.