Advertisements
Advertisements
प्रश्न
Using matrices, solve the following system of linear equations :
x + 2y − 3z = −4
2x + 3y + 2z = 2
3x − 3y − 4z = 11
Advertisements
उत्तर
The system of equations can be written in the form AX = B, where
A `= [(1,2,-3),(2,3,2),(3,-3,-4)],` X`=[("x"),("y"),("z")]` and B =`[(-4),(2),(11)]`
|A| = 1 (-12+6) - 2 (-8 - 6) - 3 (-6 - 9) = 67 ≠ 0
Therefore, A is non singular and so its inverse exists.
A11 = -6, A12 = 14, A13 = -15
A21 = 17, A22 = 5, A23 = 9
A31 = 13, A32 = -8, A33 = -1
Therefore, `"A"^-1 = 1/67[(-6,17,13),(14,5,-8),(-15,9,-1)]`
So X = A-1 B `=1/67[(-6,17,13),(14,5,-8),(-15,9,-1)][(-4),(2),(11)]`
i.e. `[("x"),("y"),("z")]=1/67[(201),(-134),(67)]=[(3),(-2),(1)]`
Hence, x = 3, y = -2 and z = 1
APPEARS IN
संबंधित प्रश्न
Let A be a square matrix of order 3 × 3, then | kA| is equal to
(A) k|A|
(B) k2 | A |
(C) k3 | A |
(D) 3k | A |
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.
Let A = [aij] be a square matrix of order 3 × 3 and Cij denote cofactor of aij in A. If |A| = 5, write the value of a31 C31 + a32 C32 a33 C33.
Which of the following is not correct?
Show that Δ = `|(x, "p", "q"),("p", x, "q"),("q", "q", x)| = (x - "p")(x^2 + "p"x - 2"q"^2)`
If x = – 4 is a root of Δ = `|(x, 2, 3),(1, x, 1),(3, 2, x)|` = 0, then find the other two roots.
If x, y ∈ R, then the determinant ∆ = `|(cosx, -sinx, 1),(sinx, cosx, 1),(cos(x + y), -sin(x + y), 0)|` lies in the interval.
The determinant ∆ = `|(sqrt(23) + sqrt(3), sqrt(5), sqrt(5)),(sqrt(15) + sqrt(46), 5, sqrt(10)),(3 + sqrt(115), sqrt(15), 5)|` is equal to ______.
The value of the determinant ∆ = `|(sin^2 23^circ, sin^2 67^circ, cos180^circ),(-sin^2 67^circ, -sin^2 23^circ, cos^2 180^circ),(cos180^circ, sin^2 23^circ, sin^2 67^circ)|` = ______.
If a1, a2, a3, ..., ar are in G.P., then prove that the determinant `|("a"_("r" + 1), "a"_("r" + 5), "a"_("r" + 9)),("a"_("r" + 7), "a"_("r" + 11), "a"_("r" + 15)),("a"_("r" + 11), "a"_("r" + 17), "a"_("r" + 21))|` is independent of r.
If A is invertible matrix of order 3 × 3, then |A–1| ______.
`|(0, xyz, x - z),(y - x, 0, y z),(z - x, z - y, 0)|` = ______.
The maximum value of `|(1, 1, 1),(1, (1 + sintheta), 1),(1, 1, 1 + costheta)|` is `1/2`
`"A" = abs ((1/"a", "a"^2, "bc"),(1/"b", "b"^2, "ac"),(1/"c", "c"^2, "ab"))` is equal to ____________.
`abs ((1 + "a", "b", "c"),("a", 1 + "b", "c"),("a", "b", 1 + "c")) =` ____________
The value of the determinant `abs ((1,0,0),(2, "cos x", "sin x"),(3, "sin x", "cos x"))` is ____________.
Find the minor of the element of the second row and third column in the following determinant `[(2,-3,5),(6,0,4),(1,5,-7)]`
If `Delta = abs((5,3,8),(2,0,1),(1,2,3)),` then write the minor of the element a23.
If `"abc" ne 0 "and" abs ((1 + "a", 1, 1),(1, 1 + "b", 1),(1,1,1 + "c")) = 0, "then" 1/"a" + 1/"b" + 1/"c" =` ____________.
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`?
Value of `|(2, 4),(-1, 2)|` is
In a third order matrix aij denotes the element of the ith row and the jth column.
A = `a_(ij) = {(0",", for, i = j),(1",", f or, i > j),(-1",", f or, i < j):}`
Assertion: Matrix ‘A’ is not invertible.
Reason: Determinant A = 0
Which of the following is correct?
