 Hi, and welcome to the session. I am Deepika here. Let's discuss a question. The question says, by using properties of determinants, show that determinant x plus 4, 2x, 2x, 2x, x plus 4, 2x, 2x, 2x, x plus 4 is equal to 5x plus 4 into 4 minus x whole square. Let us start the solution. We have on left hand side as is equal to x plus 4, 2x, 2x, 2x, 2x, x plus 4, 2x, 2x, 2x, 2x, 2x, 2x, 2x, x plus 4. By applying 1 goes to r1 plus r2 plus r3, we get r left hand side is equal to, now, x plus 4 plus 2x plus 2x, this is equal to 5x plus 4. Again, 2x plus x plus 4 plus 2x, this is again equal to 5x plus 4. Again, 2x plus 2x plus x plus 4, which is again equal to 5x plus 4. r2 is same, 2x, x plus 4, 2x, and r3 is also same, 2x, 2x, x plus 4. Now, we see that we can take 5x plus 4 common from row 1. So, by taking factor 5x plus 4 common from r1, we get r left hand side is equal to 5x plus 4 into 1, 1, 1, 2x plus 4. And this is again 2x, 2x, x plus 4. Now, by applying 2 goes to r2 minus r3, we get, our left hand side is equal to 5x plus 4 into 1, 1, 1, r2 goes to r2 minus r3, 2x minus 2x is equal to 0. x plus 4 minus 2x is equal to 4 minus x, 2x minus x minus 4, this is x minus 4, and r3 is same, 2x, 2x, x plus 4. Again, we can see that we can take common factor as 4 minus x from r2. So, by taking factor 4 minus x common from r2, we get r left hand side is equal to 5x plus 4, 4 minus x into 1, 1, 1, 0, 1 minus 1. Again, by applying c2 goes to c2 plus c3, we get, our left hand side is equal to 5x plus 4, 4 minus x. Now, c2 goes to c2 plus c3, that is 1 plus 1, 2, 1 plus minus 1, 0, 2x plus x plus 4, this is equal to 3x plus 4. And c3 is as it is determinant around r2, we get this is equal to 5x plus 4, 4 minus x into 0 minus 0 plus 0 minus of minus 1 into 1, 2, 2x, 3x plus 4. That is this we got by expanding along, this is equal to 5x plus 4, 4 minus x into 3x plus 4 minus 4x, which is again equal to 5x plus 4, 4 minus x into 4 minus x. That is equal to 5x plus 4 into 4 minus x square, this is equal to r right hand side, hence r left hand side is equal to right hand side, we have proved, I hope the question is clear to you, buy and have a nice day.