What is the value of determinant given if x, y, z are in AP with common difference d.
∣∣ ∣∣x+yy1y+zz1z+xx1∣∣ ∣∣
−3d2
You can use the property of the determinant that if the terms of one row is given as a sum of two other terms then the determinant can be written as the sum of two determinants as given below,
∣∣ ∣∣x+yy1y+zz1z+xx1∣∣ ∣∣=∣∣ ∣∣xy1yz1zx1∣∣ ∣∣+∣∣ ∣∣yy1zz1xx1∣∣ ∣∣
=∣∣ ∣∣xy1yz1zx1∣∣ ∣∣+0
R2→R2−R1
R3→R3−R1
=∣∣ ∣∣xy1y−xz−y0z−xx−y0∣∣ ∣∣
=∣∣ ∣∣xy1dd02d−d0∣∣ ∣∣
=+1(−d2−2d2)
=−3d2