Show that
`|{:(1+a^(2)-b^(2),,2ab,,-2b),(2ab,,1-a^(2)+b^(2),,2a),(2b,,-2a,,1-a^(2)-b^(2)):}| = (1+a^(2) +b^(2))^(3)`

Using the property of determinants andd without expanding in following exercises 1 to 7 prove that |{:(1+a^2-b^2,2ab,-2b),(2ab,1-a^2+b^2,2a),(2b,-2a,1-a^2-b^2):}|=(1+a^2+b^2)^3

Without expanding the determinant, prove that {:[( a, a ^(2), bc ),( b ,b ^(2) , ca),( c, c ^(2) , ab ) ]:} ={:[( 1, a^(2) , a^(3) ),( 1,b^(2) , b^(3) ),( 1, c^(2),c^(3)) ]:}

the value of the determinant |{:((a_(1)-b_(1))^(2),,(a_(1)-b_(2))^(2),,(a_(1)-b_(3))^(2),,(a_(1)-b_(4))^(2)),((a_(2)-b_(1))^(2),,(a_(2)-b_(2))^(2) ,,(a_(2)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(3)-b_(1))^(2),,(a_(3)-b_(2))^(2),,(a_(3)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(4)-b_(1))^(2),,(a_(4)-b_(2))^(2),,(a_(4)-b_(3))^(2),,(a_(4)-b_(4))^(2)):}| is

Prove that: {:|(1,a, a^2-bc), (1,b,b^2-ca),(1,c,c^2-ab)|:}=0

Factorise a ^(3) - 3a^(2) b + 3ab ^(2) -b ^(3)

Prove that |{:(a^2,bc,ac+c^2),(a^2+ab,b^2,ac),(ab,b^2+bc,c^2):}|=4a^2b^2c^2

The determinant |{:(b^2-ab,b-c,bc-ac),(ab-a^2,a-b,b^2-ab),(bc-ac,c-a,ab-a^2):}| equals …….

Compute the following : (i) [{:(a,b),(-b,a):}]+[{:(a,b),(b,a):}] (ii) [{:(a^(2)+b^(2),b^(2)+c^(2)),(a^(2)+c^(2),a^(2)+b^(2)):}]+[{:(2ab,2bc),(-2ac,-2ab):}] (iii) [{:(-1,4,-6),(8,5,16),(2,8,5):}]+[{:(12,7,6),(8,0,5),(3,2,4):}] (iv) [{:(cos^(2)x,sin^(2)x),(sin^(2)x,cos^(2)x):}]+[{:(sin^(2)x,cos^(2)x),(cos^(2)x,sin^(2)x):}]

Let Delta =|{:(-bc,b^(2)+bc,c^(2)+bc),(a^(2)+ac,-ac,c^(2)+ac),(a^(2)+ab,b^(2)+ab,-ab):}| and the equation x^(3)-px^(2)+qx-r=0 has roots a,b,c, where a,b,c in R^(+) The value of Delta is

Using the properties of determinants, prove the following |{:(a^2,b^2,c^2),((a+1)^2,(b+1)^2,(c+1)^2),((a-1)^2,(a-1)^2,(c-1)^2):}|=4|{:(a^2,b^2,c^2),(a,b,c),(1,1,1):}|