Advertisements
Advertisements
Question
Using properties of determinants prove the following: `|[1,x,x^2],[x^2,1,x],[x,x^2,1]|=(1-x^3)^2`
Solution
The given determinant is `|[1,x,x^2],[x^2,1,x],[x,x^2,1]|`
Applying the transformation c1 → c1 + c2 + c3, we get
`|[1,x,x^2],[x^2,1,x],[x,x^2,1]|=|[1+x+x^2,x,x^2],[x^2+1+x,1,x],[x+x^2+1,x^2,1]|=(1+x+x^2)|[1,x,x^2],[1,1,x],[1,x^2,1]|`
Again applying the transformation R1 → R1 − R2 and R2 → R2 − R3, we get
`(1+x+x^2)|[1,x,x^2],[1,1,x],[1,x^2,1]|=(1+x+x^2)|[0,x-1,x^2-x],[0,1-x^2,x-1],[1,x^2,1]|=(1+x+x^2)(x-1)^2|[0,1,x],[0,-x-1,1],[1,x^2,1]|`
`=(x^3-1)(x-1){0-0+(1+x+x^2)}=(x^3-1)(x-1)(x^2+x+1)`
`=(x^3-1)(x^3-1)=(x^3-1)^2=(1-x^3)^2`
hence ` |[1,x,x^2],[x^2,1,x],[x,x^2,1]|=(1-x^3)^2`
APPEARS IN
RELATED QUESTIONS
Using the property of determinants and without expanding, prove that:
`|(a-b,b-c,c-a),(b-c,c-a,a-b),(a-a,a-b,b-c)| = 0`
By using properties of determinants, show that:
`|(x+y+2z, x, y),(z, y+z+2z,y),(z,x,z+x+2y)| = 2(x+y+z)^3`
Using properties of determinants, prove that:
`|(1, 1+p, 1+p+q),(2, 3+2p, 4+3p+2q),(3,6+3p,10+6p+3q)| = 1`
Using properties of determinants, prove the following :
Using properties of determinants, prove that \[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)\] .
Using properties of determinants, prove that:
`|(a,b,b+c),(c,a,c+a),(b,c,a+b)|` = (a+b+c)(a-c)2
Using properties of determinants, prove that:
`|[a^2 + 1, ab, ac], [ba, b^2 + 1, bc ], [ca, cb, c^2+1]| = a^2 + b^2 + c^2 + 1`
Using properties of determinants, prove that:
`|[a^2 + 1, ab, ac], [ba, b^2 + 1, bc ], [ca, cb, c^2+1]| = a^2 + b^2 + c^2 + 1`
Using properties of determinant prove that
`|(b+c , a , a), (b , c+a, b), (c, c, a+b)|` = 4abc
Solve for x : `|("a"+"x","a"-"x","a"-"x"),("a"-"x","a"+"x","a"-"x"),("a"-"x","a"-"x","a"+"x")| = 0`, using properties of determinants.
Using properties of determinants, find the value of x for which
`|(4-"x",4+"x",4+"x"),(4+"x",4-"x",4+"x"),(4+"x",4+"x",4-"x")|= 0`
Without expanding evaluate the following determinant:
`|(1, "a", "b" + "c"),(1, "b", "c" + "a"),(1, "c", "a" + "b")|`
Find the value (s) of x, if `|(1, 2x, 4x),(1, 4, 16),(1, 1, 1)|` = 0
Without expanding the determinants, show that `|("b" + "c", "bc", "b"^2"c"^2),("c" + "a", "ca", "c"^2"a"^2),("a" + "b", "ab", "a"^2"b"^2)|` = 0
Using properties of determinant show that
`|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)|` = 0
If `|("x"^"k", "x"^("k" + 2), "x"^("k" + 3)),("y"^"k", "y"^("k" + 2), "y"^("k" + 3)),("z"^"k", "z"^("k" + 2), "z"^("k" + 3))|` = (x - y) (y - z) (z - x)`(1/"x"+ 1/"y" + 1/"z") ` then
Answer the following question:
Without expanding determinant show that
`|(l, "m", "n"),("e", "d", "f"),("u", "v", "w")| = |("n", "f", "w"),(l, "e", "u"),("m", "d", "v")|`
Answer the following question:
If `|("a", 1, 1),(1, "b", 1),(1, 1, "c")|` = 0 then show that `1/(1 - "a") + 1/(1 - "b") + 1/(1 - "c")` = 1
Evaluate: `|("a" + x, y, z),(x, "a" + y, z),(x, y, "a" + z)|`
Evaluate: `|(x + 4, x, x),(x, x + 4, x),(x, x, x + 4)|`
The maximum value of Δ = `|(1, 1, 1),(1, 1 + sin theta, 1),(1 + cos theta, 1, 1)|` is ______. (θ is real number)
If the ratio of the H.M. and GM. between two numbers a and bis 4 : 5, then a: b is
Let 'A' be a square matrix of order 3 × 3, then |KA| is equal to:
Without expanding determinants find the value of `|(10,57,107),(12,64,124),(15,78,153)|`
By using properties of determinant prove that `|(x + y, y+z, z +x),(z,x,y),(1,1,1)| =0`
Without expanding evaluate the following determinant:
`|(1, a, b + c), (1, b, c + a), (1, c, a + b)|`
Without expanding determinant find the value of `|(10,57,107),(12,64,124),(15,78,153)|`
By using properties of determinant prove that `|(x+y,y+z,z+x),(z,x,y),(1,1,1)|` = 0
