Home
Class 12
MATHS
By using properties of determinants. Sho...

By using properties of determinants. Show that:`|[a^2+1,a b, a c],[ a b,b^2+1,b c],[c a, c b, c^2+1]|=(1+a^2+b^2+b^2)`

Text Solution

Verified by Experts

First let us reduce the given matrix
`=(a^2+1) times[[b^2+1,bc],[cb,c^2+1]]-abtimes[[ab,bc],[ca,c^2+1]]+actimes[[ab,b^2+1],[ca,cb]]`
Now simplify the above reduced matrix
`=(a^2+1) times((b^2+1)times(c^2+1)-(bc times cb))-abtimes((abtimes(c^2+1))-(catimesbc))+actimes((abtimescb)-(catimes(b^2+1))`
...
Promotional Banner

Similar Questions

Explore conceptually related problems

3. Using properties of determinants, show that :|[b+c,a,b] , [c+a,c,a] , [a+b,b,c]| = (a + b + c) (a-c)^2

By using properties of determinants. Show that: (i) |1a a^2 1bb^2 1cc^2|=(a-b)(b-c)(c-a) (ii) |1 1 1a b c a^3b^3c^3|=(a-b)(b-c)(c-a)(a+b+c)

Using properties of determinants, prove that |[a,b,c] , [a^2,b^2,c^2] , [b+c,c+a,a+b]|=(a+b+c)(a-b)(b-c)(c-a)

Using properties of determinant show that: det[[1,a,-bc1,b,-ca1,c,-ab]]=(a-b)(b-c)(c-a)det[[1,b,-ca1,c,-ab]]=(a-b)(b-c)(c-a)

Using properties of determinants, prove that following |(a+b+2c,a,b),(c,b+c+2a,b),(c,a,c+a+2b)|=2(a+b+c)^3

5. Using the properties of determinants, prove that |[a+b,b+c,c+a] , [b+c,c+a,a+b] , [c+a,a+b,b+c]|=2|[a,b,c] , [b,c,a] , [c,a,b]|

Using properties of determinant prove that |a+b+c-c-b-c a+b+c-a-b-a a+b+c|=2(a+b)(b+c)(c+a)

Using properties of determinants Prove that |{:(a+b+c,,-c,,-b),(-c,,a+b+c,,-a),( -b,,-a,,a+b+c):}| = 2 (a+b) (b+c) (c+a)

Show that |[1,a,a^2],[1,b,b^2],[1,c,c^2]|=(a-b)(b-c)(c-a)

Show that a-b is a factor of |[1,a,a^(2)],[1,b,b^(2)],[1,c,c^(2)]|