Advertisements
Advertisements
Question
Answer the following question:
Without expanding determinant show that
`|(0, "a", "b"),(-"a", 0, "c"),(-"b", -"c", 0)|` = 0
Advertisements
Solution
Let D = `|(0, "a", "b"),(-"a", 0, "c"),(-"b", -"c", 0)|`
By taking ( –1) common from each of R1, R2, R3, we get,
D = `(-1)^3|(0, -"a", -"b"),("a", 0, -"c"),("b", "c", 0)|`
By interchanging rows and columns, we get,
D = `(-1)|(0, "a", "b"),(-"a", 0, "c"),(-"b", -"c", 0)|`
∴ D = – 1(D)
∴ 2D = 0
∴ D = 0
∴ `|(0, "a", "b"),(-"a", 0, "c"),(-"b", -"c", 0)|` = 0
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`
Using the property of determinants and without expanding, prove that:
`|(2,7,65),(3,8,75),(5,9,86)| = 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:
`|(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:
`|(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:
`|(alpha, alpha^2,beta+gamma),(beta, beta^2, gamma+alpha),(gamma, gamma^2, alpha+beta)|` = (β – γ) (γ – α) (α – β) (α + β + γ)
Using properties of determinants, prove that
`|(sin alpha, cos alpha, cos(alpha+ delta)),(sin beta, cos beta, cos (beta + delta)),(sin gamma, cos gamma, cos (gamma+ delta))| = 0`
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
If `|(4 + x, 4 - x, 4 - x),(4 - x, 4 + x, 4 - x),(4 - x, 4 - x, 4 + x)|` = 0, then find the values of x.
Without expanding determinants, show that
`|(1, 3, 6),(6, 1, 4),(3, 7, 12)| + |(2, 3, 3),(2, 1, 2),(1, 7, 6)| = 10|(1, 2, 1),(3, 1, 7),(3, 2, 6)|`
Without expanding determinants, prove that `|(1, yz, y + z),(1, zx, z + x),(1, xy, x + y)| = |(1, x, x^2),(1, y, y^2),(1, z, z^2)|`.
Find the value (s) of x, if `|(1, 2x, 4x),(1, 4, 16),(1, 1, 1)|` = 0
If `|(4 + x, 4 - x, 4 - x),(4 - x,4 + x,4 - x),(4 - x,4 - x, 4 + x)|` = 0, then find the values of x.
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
Select the correct option from the given alternatives:
The system 3x – y + 4z = 3, x + 2y – 3z = –2 and 6x + 5y + λz = –3 has at least one Solution when
Prove that: `|(y^2z^2, yz, y + z),(z^2x^2, zx, z + x),(x^2y^2, xy, x + y)|` = 0
If A + B + C = 0, then prove that `|(1, cos"c", cos"B"),(cos"C", 1, cos"A"),(cos"B", cos"A", 1)|` = 0
Find the value of θ satisfying `[(1, 1, sin3theta),(-4, 3, cos2theta),(7, -7, -2)]` = 0
The determinant `|(sin"A", cos"A", sin"A" + cos"B"),(sin"B", cos"A", sin"B" + cos"B"),(sin"C", cos"A", sin"C" + cos"B")|` is equal to zero.
The value of the determinant `abs ((alpha, beta, gamma),(alpha^2, beta^2, gamma^2),(beta + gamma, gamma + alpha, alpha + beta)) =` ____________.
If the ratio of the H.M. and GM. between two numbers a and bis 4 : 5, then a: b is
`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 ______.
Without expanding determinant 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 evaluate the following determinant.
`|(1, a, a + 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)|`
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`
By using properties of determinants, 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 evaluate the following determinant.
`|(1, a, b+c),(1, b, c+a),(1, c, a+b)|`
if `|(a, b, c),(m, n, p),(x, y, z)| = k`, then what is the value of `|(6a, 2b, 2c),(3m, n, p),(3x, y, z)|`?
Without expanding evaluate the following determinant.
`|(1, a, b + c),(1, b, c + a),(1, c, a + b)|`
