Advertisements
Advertisements
प्रश्न
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)|`.
Advertisements
उत्तर
L.H.S. = `|(1, yz, y + z),(1, zx, z + x),(1, xy, x + y)|`
= `|(1, yz, x + y + z - x),(1, zx, y + z + x - y),(1, xy, z + x + y - z)|`
= `|(1, yz, x + y + z),(1, zx, x + y + z),(1, xy, x + y + z)| - |(1, yz, x),(1, zx, y),(1, xy, z)|`
= `x + y + z |(1, yz, 1),(1, zx, 1),(1, xy, 1)| - |(1, yz, x),(1, zx, y),(1, xy, z)|`
= `(x + y + z)0 - |(1, yz, x),(1, zx, y),(1, xy, z)|`
= `0 -|(1/x xx x, 1/x xx yz, 1/x xx x xx x),(1/yxxy, 1/yxxyxxzx, 1/yxxyxxy),(1/zxxz, 1/zxxzxx xy, 1/zxxzxxz)|`
Taking `1/x, 1/y and 1/z` from R1, R2 & R3.
`-1/(xyz)|(x, xyz, x^2),(y, xyz, y^2),(z, xyz, z^2)|`
Taking xyz from C2
`-(xyz)/(xyz)|(x, 1, x^2),(y, 1, y^2),(z, 1, z^2)|=-|(x,1,x^2),(y,1,y^2),(z,1,z^2)|`
C1 ↔ C2
= `|(1,x,x^2),(1,y,y^2),(1,z,z^2)|`
APPEARS IN
संबंधित प्रश्न
Using the property of determinants and without expanding, prove that:
`|(1, bc, a(b+c)),(1, ca, b(c+a)),(1, ab, c(a+b))| = 0`
By using properties of determinants, show that:
`|(a-b-c, 2a,2a),(2b, b-c-a,2b),(2c,2c, c-a-b)| = (a + b + c)^2`
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 propertiesof determinants prove that:
`|(x , x(x^2), x+1), (y, y(y^2 + 1), y+1),( z, z(z^2 + 1) , z+1) | = (x-y) (y - z)(z - x)(x + y+ z)`
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`
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.
Without expanding evaluate the following determinant:
`|(2, 3, 4),(5, 6, 8),(6x, 9x, 12x)|`
Using properties of determinant show that
`|("a" + "b", "a", "b"),("a", "a" + "c", "c"),("b", "c", "b" + "c")|` = 4abc
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)|`
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
If cos2θ = 0, then `|(0, costheta, sin theta),(cos theta, sin theta,0),(sin theta, 0, cos theta)|^2` = ______.
The determinant `abs (("a","bc","a"("b + c")),("b","ac","b"("c + a")),("c","ab","c"("a + b"))) =` ____________
The value of the determinant `abs ((alpha, beta, gamma),(alpha^2, beta^2, gamma^2),(beta + gamma, gamma + alpha, alpha + beta)) =` ____________.
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 evaluate the following determinant.
`|(1, a, b + c),(1, b, c + a),(1, c, a + b)|`
