[b+c,a,a],[b,c+a,b],[c,c,a+b]|=

Using properties of determinants prove the following. abs[[b+c,a,a],[b,c+a,b],[c,c,a+b]]=4abc

Prove that the following. [[b+c,a,a],[b,c+a,b],[c,c,a+b]] =4ab

The product of all values of t , for which the system of equations (a-t)x+b y+c z=0,b x+(c-t)y+a z=0,c x+a y+(b-t)z=0 has non-trivial solution, is (a) |[a, -c, -b], [-c, b, -a], [-b, -a, c]| (b) |[a, b, c], [b, c, a], [c, a, b]| (c) |[a, c, b], [b, a, c], [c, b, a]| (d) |[a, a+b, b+c], [b, b+c, c+a], [c, c+a, a+b]|

The product of all values of t , for which the system of equations (a-t)x+b y+c z=0,b x+(c-t)y+a z=0,c x+a y+(b-t)z=0 has non-trivial solution, is (a) |[a, -c, -b], [-c, b, -a], [-b, -a, c]| (b) |[a, b, c], [b, c, a], [c, a, b]| (c) |[a, c, b], [b, a, c], [c, b, a]| (d) |[a, a+b, b+c], [b, b+c, c+a], [c, c+a, a+b]|

Prove that {:[( b+c,a,a), ( b,c+a,b),( c,c,b+a) ]:}

Prove that {:[( b+c,a,a), ( b,c+a,b),( c,c,b+a) ]:}

Prove that {:[( b+c,a,a), ( b,c+a,b),( c,c,b+a) ]:}

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]|

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]|

Show without expanding at any stage 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]|