 Hello and welcome to this session. Let us understand the following question which says, proof that determinant with elements a square bc ac plus c square a square plus ab b square ac ab b square plus bc c square is equal to 4a square b square c square. Now let us proceed on to the solution, considering LHS that is determinant with elements a square bc ac plus c square a square plus ab b square ac ab b square plus bc c square. Now taking common a from column c 1, b from column c 2 and c from column c 3. Taking abc common from c 1, c 2, c 3 respectively we get abc outside multiplied by determinant with elements a c a plus c, a plus b, b, a, b, b plus c, c. Now applying c 1 tends to c 1 plus c 2 minus c 3, we get column operation on column c 1. So c 1 becomes c 1 plus c 2 minus c 3, c 2 minus c 3 will give us, we can see here c minus a minus c that is minus a. So ab 1 minus a, a minus a is equal to 0. Similarly here we get a plus b plus b minus a that is 2b. Similarly here also we get 2b and rest of both the columns remain same c a plus c, b, a, b plus c, c. Now taking common 2b from c 1, we get abc multiplied by 2b multiplied by determinant 0, c, a plus c, 1, b, a, 1, b plus c, c. Now applying 2 tends to r 2 minus r 3. We will be applying the row operation on this row and subtracting so row 2 becomes row 2 minus row 3 which becomes now 2ab square c multiplied by determinant with element 0, c, a plus c. First row remains like this, second row becomes 0, b minus b plus c is equal to minus c and a minus c is a minus c and third row remains as it is. Now expanding along this column we get elevating this column and row it becomes 2ab square c multiplied by 1 multiplied by c multiplied by a minus c plus c multiplied by a plus c which is equal to 2ab square c multiplied by ac minus c square plus ac plus c square which is equal to 2ab square c multiplied by minus c square plus c square gets cancelled plus ac plus ac becomes twice of ac which is equal to 4ab square c square which is equal to right hand side. Therefore LHS is equal to RHS hence proved. I hope you understood this question. Bye and have a nice day.