Prove that: `|{:(-2a,, a+b,, a+c),( b+a,,-2b,,b+c),(c+a,, c+b,,-2c):}|=4(a+b)(b+c)(c+a)`

Prove that |{:(a-b-c, 2a, 2a), (2b, b-c-a, 2b), (2c, 2c, c-a-b):}|=(a+b+c)^(3)

Using the property of determinants prove that {:|( 3a,-a+b,-a+c),( -b+a, 3b,-b+c) ,( -c+a,-c+b,3c) |:} = 3( a+b+c) ( ab+bc+ca)

Prove that {:|( b+c,a,a), ( b,c+a,b),( c,c,b+a) |:} = 4abc

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

Prove that {:|( a,a+b,a+b+c) ,( 2a,3a+2b,4a+3b+2c),( 3a,6a+3b,10a+6b+3c)|:}=a^(3)

Prove that |(b+c,a,a),(b,c+a,b),(c,c,a+b)|= 4ac

Prove that {:[(1,ab,a+b),(1,bc,b+c),(1,ca,c+a):}]=(a-b)(b-c)(c-a)

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)) |:}

a. Minimize z =-3x+4y subject to constraints. x+2yle8 3x+2yle12 xge0, yge0 by graphical method. b. Prove that {:abs((1,a,a^2),(1,b,b^2),(1,c ,c^2)):} = (a - b)(b-c)(c-a)

Using the factor theorem it is found that b + c , c + a and a + b are three factors of the determine : |(-2a,a+b,a+c),(b+a,-2b,b+c),(c+a,c+b,-2c)| . The other factor in the value of the determine is :