Advertisements
Advertisements
Question
Using properties of determinants, prove that `|[2y,y-z-x,2y],[2z,2z,z-x-y],[x-y-z,2x,2x]|=(x+y+z)^3`
Advertisements
Solution
We need to prove that `|[2y,y-z-x,2y],[2z,2z,z-x-y],[x-y-z,2x,2x]|=(x+y+z)^3`
`|[2y,y-z-x,2y],[2z,2z,z-x-y],[x-y-z,2x,2x]|`
On applying R1→R1+R2+R3, we get
`=|[x+y+z,x+y+z,x+y+z],[2z,2z,z-x-y],[x-y-z,2x,2x]|`
Taking x+y+z common from the first row, we get
`=(x+y+z)|[1,1,1],[2z,2z,z-x-y],[x-y-z,2x,2x]|`
Now, applying C2→C2−C1 and C3→C3−C1 , we get:
`=(x+y+z)|[1,0,0],[2z,0, (z-x-y)-(2z)],[x-y-z, 2x-(x-y-z), 2x-(x-y-z)]|`
`= (xy+z) = |(1,0,0),(2z,0,-(z+x+y)),(x-y-z,(x+y+z) , x+y+z)|`
`= (x+y+z)^3 |(1,0,0),(2z,0,-1),(x-y-z , 1 , 1)|`
`= (x+y+z)^3 [1|(0,-1),(1,1)| - 0 |+0|]`
`= (x+y+z)^3 [|(0,-1),(1,1)|]`
`= (x+y+z)^3` = RHS
APPEARS IN
RELATED QUESTIONS
Using properties of determinants, prove that
`|[b+c,c+a,a+b],[q+r,r+p,p+q],[y+z,z+x,x+y]|=2|[a,b,c],[p,q,r],[x,y,z]|`
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`
Using the property of determinants and without expanding, prove that:
`|(b+c, q+r, y+z),(c+a, r+p, z +x),(a+b, p+q, x + y )| = 2|(a,p,x),(b,q,y),(c, r,z)|`
By using properties of determinants, show that:
`|(0,a, -b),(-a,0, -c),(b, c,0)| = 0`
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:
`|(1,x,x^2),(x^2,1,x),(x,x^2,1)| = (1-x^3)^2`
Using properties of determinants, prove that
`|(a^2 + 2a,2a + 1,1),(2a+1,a+2, 1),(3, 3, 1)| = (a - 1)^3`
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, show that `|("a" + "b", "a", "b"),("a", "a" + "c", "c"),("b", "c", "b" + "c")|` = 4abc.
Find the value (s) of x, if `|(1, 2x, 4x),(1, 4, 16),(1, 1, 1)|` = 0
By using properties of determinants, prove that `|(x + y, y + z, z + x),(z, x, y),(1, 1, 1)|` = 0.
Prove that `|(x + y, y + z, z + x),(z + x, x + y, y + z),(y + z, z + x, x + y)| = 2|(x, y, z),(z, x, y),(y, z, x)|`
Select the correct option from the given alternatives:
Let D = `|(sintheta*cosphi, sintheta*sinphi, costheta),(costheta*cosphi, costheta*sinphi, -sintheta),(-sintheta*sinphi, sintheta*cosphi, 0)|` then
Select the correct option from the given alternatives:
Which of the following is correct
Answer the following question:
Without expanding determinant show that
`|("b" + "c", "bc", "b"^2"c"^2),("c" + "a", "ca", "c"^2"a"^2),("a" + "b", "ab", "a"^2"b"^2)|` = 0
Answer the following question:
Without expanding determinant show that
`|(0, "a", "b"),(-"a", 0, "c"),(-"b", -"c", 0)|` = 0
Evaluate: `|(x + 4, x, x),(x, x + 4, x),(x, x, x + 4)|`
Find the value of θ satisfying `[(1, 1, sin3theta),(-4, 3, cos2theta),(7, -7, -2)]` = 0
If `[(4 - x, 4 + x, 4 + x),(4 + x, 4 - x, 4 + x),(4 + x, 4 + x, 4 - x)]` = 0, then find values of x.
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 ______.
Let Δ = `|("a", "p", x),("b", "q", y),("c", "r", z)|` = 16, then Δ1 = `|("p" + x, "a" + x, "a" + "p"),("q" + y, "b" + y, "b" + "q"),("r" + z, "c" + z, "c" + "r")|` = 32.
Let P be any non-empty set containing p elements. Then, what is the number of relations on P?
`f : {1, 2, 3) -> {4, 5}` is not a function, if it is defined by which of the following?
The value of the determinant `|(1, cos(β - α), cos(γ - α)),(cos(α - β), 1, cos(γ - β)),(cos(α - γ), cos(β - γ), 1)|` is equal to ______.
If `|(α, 3, 4),(1, 2, 1),(1, 4, 1)|` = 0, then the value of α is ______.
Without expanding evaluate the following determinant.
`|(1,"a","b+c"),(1,"b","c+a"),(1,"c","a+b")|`
Without expanding the determinant, find the value of `|(10,57,107),(12,64,124),(15,78,153)|`
The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of the determinant of C2?
Without expanding determinant find the value of `|(10, 57, 107),(12, 64, 124),(15, 78, 153)|`
