Question

please explain me this answer ​

Answer 1

Answer:

prolly not the best person to explain this but imma try

so here in the first step they performed an operation on the first column c1 with was adding all 3 columns i.e c1+c2+c3

let's solve it fr to understand

a+b+2c+a+b a b

c+b+c+2a+b b+c+2a b

c+a+c+a+2b a c+a+2b

2a+2b+2c a b

2a+2b+2c b+c+2a b

2a+2b+2c a c+a+2b

2(a+b+c) a b

2(a+b+c) b+c+2a b

2(a+b+c) a c+a+2b

hopefully this made sense

next they took out 2(a+b+c) outside because it was common in all the three rows, resulting in the second determinant they've shown

further row operations are performed, lemme elaborate one to get a better idea

here R1=R1-R2 , performing the operation on row 1 leaving the other rows untouched we get,

∆= 2(a+b+c) | 1-1 a-(b+c+2a) b-b |

| 1 b+c+2a b |

| 1 a c+a+2b |

∆=2(a+b+c) | 0 a-b-c-2a 0 |

| 1 b+c+2a b |

| 1 a c+a+2b |

∆=2(a+b+c) | 0 -b-c-a 0 |

| 1 b+c+2a b |

| 1 a c+a+2b |

taking -ve common,

∆=2(a+b+c) | 0 -(a+b+c) 0 |

| 1 b+c+2a b |

| 1 a c+a+2b |

similarly perform operation on row2

once done they've expanded it like any other determinant along c1, i.e by eliminating the row and column along c1 and cross multiplying inside(idk how to explain any further ಥ‿ಥ)

expand expand expand and bam you get the answer

hopefully this made sense feel free to correct me if I went wrong somewhere [sorry I got lazy at the end(〒﹏〒)]

if you made it this far, have a nice day<3