Basic Properties of Determinants

Examples Problems on Properties of Determinants

Some important example on properties of determinants are given below:

1. Using Properties of Determinant, Prove That

∣∣∣∣xyzyzxzxy∣∣∣∣|xyzyzxzxy|

= (x + y + z)(xy + yz + zx - x² - y² - z²)

Solution:  With the help of the invariance and scalar multiple properties of determinant we can prove the above- given determinant.

Δ =

∣∣∣∣xyzyzxzxy∣∣∣∣|xyzyzxzxy|

=

∣∣∣∣x+y+zy+z+xz+x+yyzzzxy∣∣∣∣|x+y+zyzy+z+xzxz+x+yzy|

OperatingC\[1\]⟶C\[1\]+C\[2\]+C\[3\]OperatingC\[1\]⟶C\[1\]+C\[2\]+C\[3\]

=  (x + y + z)

∣∣∣∣111yzx0xy∣∣∣∣|1y01zx1xy|

= (x + y + z)

∣∣∣∣101yz−yx−yzx−zy−z∣∣∣∣|1yz0z−yx−z1x−yy−z|

Operating(R\[2\]⟶R\[2\]−R\[1\]and(R\[3\]⟶R\[3\]−R\[1\])Operating(R\[2\]⟶R\[2\]−R\[1\]and(R\[3\]⟶R\[3\]−R\[1\])

= ( x + y + z) [(z - y)(y - z)- (x - y) (x - z )

= ( x + y + z) (xy + yz + zx - x² - y² - Z²)