using properties of determinant , prove that |1+x 1 1 - Maths - Determinants

Question

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using properties of determinant , prove that
|1+x 1 1 |
|1 1+y 1 | = xyz+xy+yz+zx
|1 1 1+z|

Neha Sethi · Accepted Answer

Dear student

1+x1111+y1111+zlet △=1+x1111+y1111+zTaking x,y,z common from C1,C2,C3 respectively, we obtain△=xyz1x+11y1z1x1y+11z1x1y1z+1△=xyz 1+1x+1y+1z1y1z1+1x+1y+1z1y+11z1+1x+1y+1z1y1z+1             applying C1→C1+C2+C3⇒△=xyz1+1x+1y+1z11y1z11y+11z11y1z+1        taking 1+1x+1y+1z common from C1  ⇒△=xyz1+1x+1y+1z11y1z010001       Applying R2→R2-R1, R3→R3-R1⇒△=xyz1+1x+1y+1z11001           expanding along C1⇒△=xyz1+1x+1y+1z⇒△=xyz+yz+xz+xy
Regards

​ using properties of determinant , prove that |1+x 1 1 | |1 1+y 1 | = xyz+xy+yz+zx |1 1 1+z|

using properties of determinant , prove that

|1+x 1 1 |

|1 1+y 1 | = xyz+xy+yz+zx

|1 1 1+z|