Advertisements
Advertisements
Question
By using properties of determinants, show that:
`|(0,a, -b),(-a,0, -c),(b, c,0)| = 0`
Solution
We have,
Here, the two rows R1 and R3 are identical.
∴Δ = 0.
APPEARS IN
RELATED QUESTIONS
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 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]|`
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:
`|(y+k,y, y),(y, y+k, y),(y, y, y+k)| = k^2(3y + k)`
Evaluate `|(x, y, x+y),(y, x+y, x),(x+y, x, y)|`
Using properties of determinants, prove that:
`|(x, x^2, 1+px^3),(y, y^2, 1+py^3),(z, z^2, 1+pz^2)|` = (1 + pxyz) (x – y) (y – z) (z – x), where p is any scalar.
Using properties of determinants, prove that:
`|(3a, -a+b, -a+c),(-b+a, 3b, -b+c),(-c+a, -c+b, 3c)|`= 3(a + b + c) (ab + bc + ca)
Using properties of determinants, prove that:
`|(1+a^2-b^2, 2ab, -2b),(2ab, 1-a^2+b^2, 2a),(2b, -2a, 1-a^2-b^2)| = (1 + a^2 + b^2)^3`
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.
Using properties of determinants, find the value of x for which
`|(4-"x",4+"x",4+"x"),(4+"x",4-"x",4+"x"),(4+"x",4+"x",4-"x")|= 0`
Without expanding evaluate the following determinant:
`|(1, "a", "b" + "c"),(1, "b", "c" + "a"),(1, "c", "a" + "b")|`
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 the determinant, find the value of `|(10, 57, 107),(12, 64, 124),(15, 78, 153)|`.
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
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
`|(x"a", y"b", z"c"),("a"^2, "b"^2, "c"^2),(1, 1, 1)| = |(x, y, z),("a", "b", "c"),("bc", "ca", "ab")|`
Evaluate: `|("a" + x, y, z),(x, "a" + y, z),(x, y, "a" + z)|`
Prove that: `|(y + z, z, y),(z, z + x, x),(y, x, x + y)|` = 4xyz
The maximum value of Δ = `|(1, 1, 1),(1, 1 + sin theta, 1),(1 + cos theta, 1, 1)|` is ______. (θ is real number)
The value of the determinant `|(x , x + y, x + 2y),(x + 2y, x, x + y),(x + y, x + 2y, x)|` is ______.
If the value of a third order determinant is 12, then the value of the determinant formed by replacing each element by its co-factor will be 144.
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.
If a, b, c are the roots of the equation x3 - 3x2 + 3x + 7 = 0, then the value of `abs((2 "bc - a"^2, "c"^2, "b"^2),("c"^2, 2 "ac - b"^2, "a"^2),("b"^2, "a"^2, 2 "ab - c"^2))` is ____________.
Using properties of determinants `abs ((1, "a", "a"^2 - "bc"),(1, "b", "b"^2 - "ca"),(1, "c", "c"^2 - "ab")) =` ____________.
In a third order matrix B, bij denotes the element in the ith row and jth column. If
bij = 0 for i = j
= 1 for > j
= – 1 for i < j
Then the matrix is
Let 'A' be a square matrix of order 3 × 3, then |KA| is equal to:
If f(α) = `[(cosα, -sinα, 0),(sinα, cosα, 0),(0, 0, 1)]`, prove that f(α) . f(– β) = f(α – β).
Without expanding determinant 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)|`
By using properties of determinant prove that
`|(x+y,y+z,z+x),(z,x,y),(1,1,1)|=0`
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)|`
Without expanding determinants, find the value of `|(10, 57, 107),(12, 64, 124),(15, 78, 153)|`