Advertisements
Advertisements
Question
`|(0, xyz, x - z),(y - x, 0, y z),(z - x, z - y, 0)|` = ______.
Solution
`|(0, xyz, x - z),(y - x, 0, y z),(z - x, z - y, 0)|` = (y – z)(z – x)(y – x + xyz).
Explanation:
Let Δ = `|(0, xyz, x - z),(y - x, 0, y z),(z - x, z - y, 0)|`
C1 → C1 – C3
= `|(z - x, xyz, x - z),(z - x, 0, y - z),(z - x, z - y, 0)|`
Taking (z – x) common from C1
= `(z - x) |(1, xyz, x - z),(1, 0, y - z),(1, z - y, 0)|`
R1 → R1 – R2, R2 → R2 – R3
= `(z - x) |(0, xyz,y),(0, y - x, y - z),(1, z - y, 0)|`
Taking (y – z) common from R2
= `(z - x)(y - z) |(0, xyz, x - y),(0, 1, 1),(1, z - y, 0)|`
Expanding along C1
= `(z - x)(y - z) [1|(xyz, x - y),(1, 1)|]`
= (z – x)(y – z)(xyz – x + y)
= (y – z)(z – x)(y – x + xyz)
APPEARS IN
RELATED QUESTIONS
Find values of x, if `|[2,3],[4,5]|=|[x,3],[2x,5]|`
Without expanding at any stage, find the value of:
`|(a,b,c),(a+2x,b+2y,c+2z),(x,y,z)|`
On expanding by first row, the value of the determinant of 3 × 3 square matrix
\[A = \left[ a_{ij} \right]\text{ is }a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}\] , where [Cij] is the cofactor of aij in A. Write the expression for its value on expanding by second column.
A matrix of order 3 × 3 has determinant 2. What is the value of |A (3I)|, where I is the identity matrix of order 3 × 3.
If A is a 3 × 3 matrix, \[\left| A \right| \neq 0\text{ and }\left| 3A \right| = k\left| A \right|\] then write the value of k.
Which of the following is not correct in a given determinant of A, where A = [aij]3×3.
If x = – 4 is a root of Δ = `|(x, 2, 3),(1, x, 1),(3, 2, x)|` = 0, then find the other two roots.
The determinant ∆ = `|(cos(x + y), -sin(x + y), cos2y),(sinx, cosx, siny),(-cosx, sinx, cosy)|` is independent of x only.
If a + b + c ≠ 0 and `|("a", "b","c"),("b", "c", "a"),("c", "a", "b")|` 0, then prove that a = b = c.
If x + y + z = 0, prove that `|(x"a", y"b", z"c"),(y"c", z"a", x"b"),(z"b", x"c", y"a")| = xyz|("a", "b", "c"),("c", "a", "b"),("b", "c", "a")|`
Let f(t) = `|(cos"t","t", 1),(2sin"t", "t", 2"t"),(sin"t", "t", "t")|`, then `lim_("t" - 0) ("f"("t"))/"t"^2` is equal to ______.
If f(x) = `|(0, x - "a", x - "b"),(x + "b", 0, x - "c"),(x + "b", x + "c", 0)|`, then ______.
There are two values of a which makes determinant, ∆ = `|(1, -2, 5),(2, "a", -1),(0, 4, 2"a")|` = 86, then sum of these number is ______.
If A is invertible matrix of order 3 × 3, then |A–1| ______.
If A is a matrix of order 3 × 3, then (A2)–1 = ______.
If f(x) = `|((1 + x)^17, (1 + x)^19, (1 + x)^23),((1 + x)^23, (1 + x)^29, (1 + x)^34),((1 +x)^41, (1 +x)^43, (1 + x)^47)|` = A + Bx + Cx2 + ..., then A = ______.
If A and B are matrices of order 3 and |A| = 5, |B| = 3, then |3AB| = 27 × 5 × 3 = 405.
`"A" = abs ((1/"a", "a"^2, "bc"),(1/"b", "b"^2, "ac"),(1/"c", "c"^2, "ab"))` is equal to ____________.
If `Delta = abs((5,3,8),(2,0,1),(1,2,3)),` then write the minor of the element a23.
Let A be a square matrix of order 2 x 2, then `abs("KA")` is equal to ____________.
Find the 5th term of expansion of `(x^2 + 1/x)^10`?
For positive numbers x, y, z, the numerical value of the determinant `|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)|` is
Value of `|(2, 4),(-1, 2)|` is
