Advertisements
Advertisements
प्रश्न
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)|`
Advertisements
उत्तर
L.H.S. = `|(1, 3, 6),(6, 1, 4),(3, 7, 12)| + 4 |(2, 3, 3),(2, 1, 2),(1, 7, 6)|`
In 1st determinant, taking 2 common from C3, we get
L.H.S. = `2|(1,3,3),(6,1,2),(3,7,6)| + 4|(2,3,3),(2,1,2),(1,7,6)|`
= `|(2, 3, 3),(12, 1, 2),(6, 7, 6)| + |(8, 3, 3),(8, 1, 2),(4, 7, 6)|`
= `|(2 + 8, 3, 3),(12 + 8, 1, 2),(6 + 4, 7, 6)|`
= `|(10, 3, 3),(20, 1, 2),(10, 7, 6)|`
Interchanging rows and columns, we get
L.H.S. = `|(10, 20, 10),(3, 1, 7),(3, 2, 6)|`
Taking 10 common from R1, we get
L.H.S. = `10|(1, 2, 1),(3, 1, 7),(3, 2, 6)|`
= R.H.S.
APPEARS IN
संबंधित प्रश्न
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)
By using properties of determinants, show that:
`|(1+a^2-b^2, 2ab, -2b),(2ab, 1-a^+b^2, 2a),(2b, -2a, 1-a^2-b^2)| = (1+a^2+b^2)`
Using properties of determinants, prove the following :
Using properties of determinants, prove that:
`|(a,b,b+c),(c,a,c+a),(b,c,a+b)|` = (a+b+c)(a-c)2
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`.
Without expanding determinants, find the value of `|(2014, 2017, 1),(2020, 2023, 1),(2023, 2026, 1)|`
Find the value (s) of x, if `|(1, 4, 20),(1, -2, -5),(1, 2x, 5x^2)|` = 0
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
Using properties of determinant show that
`|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)|` = 0
Evaluate: `|(x + 4, x, x),(x, x + 4, x),(x, x, x + 4)|`
Prove that: `|(y^2z^2, yz, y + z),(z^2x^2, zx, z + x),(x^2y^2, xy, x + y)|` = 0
If cos2θ = 0, then `|(0, costheta, sin theta),(cos theta, sin theta,0),(sin theta, 0, cos theta)|^2` = ______.
Let P be any non-empty set containing p elements. Then, what is the number of relations on P?
The A.M., H.M. and G.M. between two numbers are `144/15`, 15 and 12, but not necessarily in this order then, H.M., G.M. and A.M. respectively are
In a triangle the length of the two larger sides are 10 and 9, respectively. If the angles are in A.P., then the length of the third side can be ______.
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)|`
By using properties of determinant prove that `|(x+y,y+z,z+x),(z,x,y),(1,1,1)|=0`
Without expanding determinant find the value of `|(10,57,107),(12,64,124),(15,78,153)|`
