Advertisements
Advertisements
प्रश्न
The system of equation x + y + z = 2, 3x − y + 2z = 6 and 3x + y + z = −18 has
विकल्प
a unique solution
no solution
an infinite number of solutions
zero solution as the only solution
Advertisements
उत्तर
(a) a unique solution
The given system of equations can be written in matrix form as follows:
\[\begin{bmatrix}1 & 1 & 1 \\ 3 & - 1 & 2 \\ 3 & 1 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}2 \\ 6 \\ - 18\end{bmatrix}\]
\[AX = B \]
Here,
\[A = \begin{bmatrix}1 & 1 & 1 \\ 3 & - 1 & 2 \\ 3 & 1 & 1\end{bmatrix}, X = \begin{bmatrix}x \\ y \\ z\end{bmatrix}\text{ and }B = \begin{bmatrix}2 \\ 6 \\ - 18\end{bmatrix}\]
\[\left| A \right|=1 \left( - 1 - 2 \right) - 1\left( 3 - 6 \right) + 1\left( 3 + 3 \right)\]
\[ = - 3 + 3 + 6\]
\[ = 6\neq0\]
So, the given system of equations has a unique solution.
APPEARS IN
संबंधित प्रश्न
Solve the system of linear equations using the matrix method.
5x + 2y = 4
7x + 3y = 5
Solve the system of linear equations using the matrix method.
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\cos\left( x + y \right) & - \sin\left( x + y \right) & \cos2y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1\end{vmatrix}, where A, B, C \text{ are the angles of }∆ ABC .\]
\[If ∆ = \begin{vmatrix}1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2\end{vmatrix}, ∆_1 = \begin{vmatrix}1 & 1 & 1 \\ yz & zx & xy \\ x & y & z\end{vmatrix},\text{ then prove that }∆ + ∆_1 = 0 .\]
Prove that
\[\begin{vmatrix}- bc & b^2 + bc & c^2 + bc \\ a^2 + ac & - ac & c^2 + ac \\ a^2 + ab & b^2 + ab & - ab\end{vmatrix} = \left( ab + bc + ca \right)^3\]
Solve the following determinant equation:
If \[a, b\] and c are all non-zero and
Find the area of the triangle with vertice at the point:
(2, 7), (1, 1) and (10, 8)
Find the area of the triangle with vertice at the point:
(0, 0), (6, 0) and (4, 3)
Using determinants show that the following points are collinear:
(5, 5), (−5, 1) and (10, 7)
If the points (x, −2), (5, 2), (8, 8) are collinear, find x using determinants.
Prove that :
6x + y − 3z = 5
x + 3y − 2z = 5
2x + y + 4z = 8
x − y + z = 3
2x + y − z = 2
− x − 2y + 2z = 1
If a, b, c are non-zero real numbers and if the system of equations
(a − 1) x = y + z
(b − 1) y = z + x
(c − 1) z = x + y
has a non-trivial solution, then prove that ab + bc + ca = abc.
Find the value of the determinant
\[\begin{bmatrix}4200 & 4201 \\ 4205 & 4203\end{bmatrix}\]
Write the value of the determinant \[\begin{vmatrix}2 & - 3 & 5 \\ 4 & - 6 & 10 \\ 6 & - 9 & 15\end{vmatrix} .\]
Write the value of \[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix} .\]
If ω is a non-real cube root of unity and n is not a multiple of 3, then \[∆ = \begin{vmatrix}1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1\end{vmatrix}\]
If \[A + B + C = \pi\], then the value of \[\begin{vmatrix}\sin \left( A + B + C \right) & \sin \left( A + C \right) & \cos C \\ - \sin B & 0 & \tan A \\ \cos \left( A + B \right) & \tan \left( B + C \right) & 0\end{vmatrix}\] is equal to
Solve the following system of equations by matrix method:
3x + 4y − 5 = 0
x − y + 3 = 0
Show that the following systems of linear equations is consistent and also find their solutions:
6x + 4y = 2
9x + 6y = 3
Show that each one of the following systems of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5
If A = `[(1, 2, 0), (-2, -1, -2), (0, -1, 1)]`, find A−1. Using A−1, solve the system of linear equations x − 2y = 10, 2x − y − z = 8, −2y + z = 7.
The prices of three commodities P, Q and R are Rs x, y and z per unit respectively. A purchases 4 units of R and sells 3 units of P and 5 units of Q. B purchases 3 units of Q and sells 2 units of P and 1 unit of R. Cpurchases 1 unit of P and sells 4 units of Q and 6 units of R. In the process A, B and C earn Rs 6000, Rs 5000 and Rs 13000 respectively. If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method.
2x − y + 2z = 0
5x + 3y − z = 0
x + 5y − 5z = 0
x + y = 1
x + z = − 6
x − y − 2z = 3
If A = `[[1,1,1],[0,1,3],[1,-2,1]]` , find A-1Hence, solve the system of equations:
x +y + z = 6
y + 3z = 11
and x -2y +z = 0
Transform `[(1, 2, 4),(3, -1, 5),(2, 4, 6)]` into an upper triangular matrix by using suitable row transformations
Three chairs and two tables cost ₹ 1850. Five chairs and three tables cost ₹2850. Find the cost of four chairs and one table by using matrices
Solve the following equations by using inversion method.
x + y + z = −1, x − y + z = 2 and x + y − z = 3
If A = `[(1, -1, 2),(3, 0, -2),(1, 0, 3)]`, verify that A(adj A) = (adj A)A
If `|(2x, 5),(8, x)| = |(6, 5),(8, 3)|`, then find x
`abs (("a"^2, 2"ab", "b"^2),("b"^2, "a"^2, 2"ab"),(2"ab", "b"^2, "a"^2))` is equal to ____________.
A set of linear equations is represented by the matrix equation Ax = b. The necessary condition for the existence of a solution for this system is
The system of simultaneous linear equations kx + 2y – z = 1, (k – 1)y – 2z = 2 and (k + 2)z = 3 have a unique solution if k equals:
