Advertisements
Advertisements
Question
By using properties of determinants, show that:
`|(1,1,1),(a,b,c),(a^3, b^3,c^3)|` = (a-b)(b-c)(c-a)(a+b+c)
Advertisements
Solution

APPEARS IN
RELATED QUESTIONS
Using properties of determinants, prove that
`|((x+y)^2,zx,zy),(zx,(z+y)^2,xy),(zy,xy,(z+x)^2)|=2xyz(x+y+z)^3`
By using properties of determinants, show that:
`|(1,a,a^2),(1,b,b^2),(1,c,c^2)| = (a - b)(b-c)(c-a)`
By using properties of determinants, show that:
`|(x,x^2,yz),(y,y^2,zx),(z,z^2,xy)| = (x-y)(y-z)(z-x)(xy+yz+zx)`
By using properties of determinants, show that:
`|(1,x,x^2),(x^2,1,x),(x,x^2,1)| = (1-x^3)^2`
Using properties 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)
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 show that
`[[1,1,1+x],[1,1+y,1],[1+z,1,1]] = xyz+ yz +zx+xy.`
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 determinant prove that
`|(b+c , a , a), (b , c+a, b), (c, c, a+b)|` = 4abc
Using properties of determinants, prove the following:
`|(a, b,c),(a-b, b-c, c-a),(b+c, c+a, a+b)| = a^3 + b^3 + c^3 - 3abc`.
Find the value (s) of x, if `|(1, 4, 20),(1, -2, -5),(1, 2x, 5x^2)|` = 0
By using properties of determinants, prove that `|(x + y, y + z, z + x),(z, x, y),(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
Without expanding the determinants, show that `|(l, "m", "n"),("e", "d", "f"),("u", "v", "w")| = |("n", "f", "w"),(l, "e", "u"),("m", "d", "v")|`
Without expanding evaluate the following determinant:
`|(2, 3, 4),(5, 6, 8),(6x, 9x, 12x)|`
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: `|(x + 4, x, x),(x, x + 4, x),(x, x, x + 4)|`
Prove that: `|(y + z, z, y),(z, z + x, x),(y, x, x + y)|` = 4xyz
Find the value of θ satisfying `[(1, 1, sin3theta),(-4, 3, cos2theta),(7, -7, -2)]` = 0
The value of determinant `|("a" - "b", "b" + "c", "a"),("b" - "a", "c" + "a", "b"),("c" - "a", "a" + "b", "c")|` is ______.
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 ______.
The number of distinct real roots of `|(sinx, cosx, cosx),(cosx, sinx, cosx),(cosx, cosx, sinx)|` = 0 in the interval `pi/4 x ≤ pi/4` is ______.
If A, B and C are angles of a triangle, then the determinant `|(-1, cos"C", cos"B"),(cos"C", -1, cos"A"),(cos"B", cos"A", -1)|` is equal to ______.
If cos2θ = 0, then `|(0, costheta, sin theta),(cos theta, sin theta,0),(sin theta, 0, cos theta)|^2` = ______.
If the value of a third order determinant is 12, then the value of the determinant formed by replacing each element by its co-factor will be 144.
If the determinant `|(x + "a", "p" + "u", "l" + "f"),("y" + "b", "q" + "v", "m" + "g"),("z" + "c", "r" + "w", "n" + "h")|` splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K is 8.
If `abs ((2"x",5),(8, "x")) = abs ((6,-2),(7,3)),` then the value of x is ____________.
Let 'A' be a square matrix of order 3 × 3, then |KA| is equal to:
If f(α) = `[(cosα, -sinα, 0),(sinα, cosα, 0),(0, 0, 1)]`, prove that f(α) . f(– β) = f(α – β).
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 determinants find the value of `|(10, 57, 107),(12, 64, 124),(15, 78, 153)|`
Without expanding determinants, find the value of `|(10, 57, 107), (12, 64, 124), (15, 78, 153)|`
Without expanding the determinant, find the value of `|(10,57,107),(12,64,124),(15,78,153)|`
Without expanding evaluate the following determinant.
`|(1, a, b+c),(1, b, c+a),(1, c, a+b)|`
Without expanding evaluate the following determinant.
`|(1, a, b+c), (1, b, c+a), (1, c, a+b)|`
Without expanding determinants, find the value of `|(10, 57, 107),(12, 64, 124),(15, 78, 153)|`
