Advertisements
Advertisements
प्रश्न
Without expanding evaluate the following determinant:
`|(1, "a", "b" + "c"),(1, "b", "c" + "a"),(1, "c", "a" + "b")|`
Advertisements
उत्तर
Let D = `|(1, "a", "b" + "c"),(1, "b", "c" + "a"),(1, "c", "a" + "b")|`
Applying C3 → C3 + C2, we get
D = `|(1, "a", "a" + "b" + "c"),(1, "b", "a" + "b" + "c"),(1, "c", "a" + "b" + "c")|`
Taking (a + b + c) common from C3, we get
D = `("a" + "b" + "c")|(1, "a", 1),(1, "b", 1),(1, "c", 1)|`
∴ D = (a + b + c)(0) ...[∵ C1 and C3 are identical]
∴ D = 0
APPEARS IN
संबंधित प्रश्न
Using properties of determinants prove the following: `|[1,x,x^2],[x^2,1,x],[x,x^2,1]|=(1-x^3)^2`
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:
`|(1, bc, a(b+c)),(1, ca, b(c+a)),(1, ab, c(a+b))| = 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+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
`|(a^2 + 2a,2a + 1,1),(2a+1,a+2, 1),(3, 3, 1)| = (a - 1)^3`
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,b,b+c),(c,a,c+a),(b,c,a+b)|` = (a+b+c)(a-c)2
Without expanding evaluate the following determinant:
`|(2, 3, 4),(5, 6, 8),(6x, 9x, 12x)|`
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)|`
Without expanding determinants show that
`|(1, 3, 6),(6, 1, 4),(3, 7, 12)| + 4|(2, 3, 3),(2, 1, 2),(1, 7, 6)| = 10|(1, 2, 1),(3, 1, 7),(3, 2, 6)|`
Select the correct option from the given alternatives:
The determinant D = `|("a", "b", "a" + "b"),("b", "c", "b" + "c"),("a" + "b", "b" + "c", 0)|` = 0 if
Select the correct option from the given alternatives:
Which of the following is correct
Answer the following question:
Evaluate `|(101, 102, 103),(106, 107, 108),(1, 2, 3)|` by using properties
Answer the following question:
Without expanding determinant show that
`|(x"a", y"b", z"c"),("a"^2, "b"^2, "c"^2),(1, 1, 1)| = |(x, y, z),("a", "b", "c"),("bc", "ca", "ab")|`
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 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 ______.
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)|`
