Advertisements
Advertisements
Question
x − y + 3z = 6
x + 3y − 3z = − 4
5x + 3y + 3z = 10
Advertisements
Solution
Using the equations, we get
\[D = \begin{vmatrix}1 & - 1 & 3 \\ 1 & 3 & - 3 \\ 5 & 3 & 3\end{vmatrix} = 1(9 + 9) + 1(3 + 15) + 3(3 - 15)\]
\[ = 18 + 18 - 36 = 0\]
\[ D_1 = \begin{vmatrix}6 & - 1 & 3 \\ - 4 & 3 & - 3 \\ 10 & 3 & 3\end{vmatrix} = 6(9 + 9) + 1( - 12 + 30) + 3( - 12 - 30)\]
\[ = 108 + 18 - 126 = 0\]
\[ D_2 = \begin{vmatrix}1 & 6 & 3 \\ 1 & - 4 & - 3 \\ 5 & 10 & 3\end{vmatrix} = 1( - 12 + 30) - 6(3 + 15) + 3(10 + 20)\]
\[ = 18 - 108 + 90 = 0\]
\[ D_3 = \begin{vmatrix}1 & - 1 & 6 \\ 1 & 3 & - 4 \\ 5 & 3 & 10\end{vmatrix} = 1(30 + 12) + 1(10 + 20) + 6(3 - 15)\]
\[ = 42 + 30 - 72 = 0\]
\[ \therefore D = D_1 = D_2 = D_3 = 0\]
Hence, the system of equations has infinitely many solutions.
APPEARS IN
RELATED QUESTIONS
Examine the consistency of the system of equations.
x + 3y = 5
2x + 6y = 8
Find the integral value of x, if \[\begin{vmatrix}x^2 & x & 1 \\ 0 & 2 & 1 \\ 3 & 1 & 4\end{vmatrix} = 28 .\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\sin^2 23^\circ & \sin^2 67^\circ & \cos180^\circ \\ - \sin^2 67^\circ & - \sin^2 23^\circ & \cos^2 180^\circ \\ \cos180^\circ & \sin^2 23^\circ & \sin^2 67^\circ\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\cos\left( x + y \right) & - \sin\left( x + y \right) & \cos2y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}a & b + c & a^2 \\ b & c + a & b^2 \\ c & a + b & c^2\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}x + \lambda & x & x \\ x & x + \lambda & x \\ x & x & x + \lambda\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}0 & x y^2 & x z^2 \\ x^2 y & 0 & y z^2 \\ x^2 z & z y^2 & 0\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]
Using determinants show that the following points are collinear:
(1, −1), (2, 1) and (4, 5)
Find values of k, if area of triangle is 4 square units whose vertices are
(−2, 0), (0, 4), (0, k)
Prove that :
5x + 7y = − 2
4x + 6y = − 3
x + y + z + 1 = 0
ax + by + cz + d = 0
a2x + b2y + x2z + d2 = 0
x + 2y = 5
3x + 6y = 15
Find the value of the determinant
\[\begin{bmatrix}4200 & 4201 \\ 4205 & 4203\end{bmatrix}\]
If \[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}\] , write the value of x.
Let \[\begin{vmatrix}x & 2 & x \\ x^2 & x & 6 \\ x & x & 6\end{vmatrix} = a x^4 + b x^3 + c x^2 + dx + e\]
Then, the value of \[5a + 4b + 3c + 2d + e\] is equal to
The value of \[\begin{vmatrix}5^2 & 5^3 & 5^4 \\ 5^3 & 5^4 & 5^5 \\ 5^4 & 5^5 & 5^6\end{vmatrix}\]
Let \[A = \begin{bmatrix}1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1\end{bmatrix},\text{ where 0 }\leq \theta \leq 2\pi . \text{ Then,}\]
The determinant \[\begin{vmatrix}b^2 - ab & b - c & bc - ac \\ ab - a^2 & a - b & b^2 - ab \\ bc - ca & c - a & ab - a^2\end{vmatrix}\]
Solve the following system of equations by matrix method:
5x + 3y + z = 16
2x + y + 3z = 19
x + 2y + 4z = 25
Solve the following system of equations by matrix method:
3x + 4y + 2z = 8
2y − 3z = 3
x − 2y + 6z = −2
Solve the following system of equations by matrix method:
2x + 6y = 2
3x − z = −8
2x − y + z = −3
Solve the following system of equations by matrix method:
Show that each one of the following systems of linear equation is inconsistent:
2x + 5y = 7
6x + 15y = 13
2x − y + 2z = 0
5x + 3y − z = 0
x + 5y − 5z = 0
The system of equation x + y + z = 2, 3x − y + 2z = 6 and 3x + y + z = −18 has
The system of linear equations:
x + y + z = 2
2x + y − z = 3
3x + 2y + kz = 4 has a unique solution if
The system of equations:
x + y + z = 5
x + 2y + 3z = 9
x + 3y + λz = µ
has a unique solution, if
(a) λ = 5, µ = 13
(b) λ ≠ 5
(c) λ = 5, µ ≠ 13
(d) µ ≠ 13
Show that if the determinant ∆ = `|(3, -2, sin3theta),(-7, 8, cos2theta),(-11, 14, 2)|` = 0, then sinθ = 0 or `1/2`.
`abs (("a"^2, 2"ab", "b"^2),("b"^2, "a"^2, 2"ab"),(2"ab", "b"^2, "a"^2))` is equal to ____________.
Choose the correct option:
If a, b, c are in A.P. then the determinant `[(x + 2, x + 3, x + 2a),(x + 3, x + 4, x + 2b),(x + 4, x + 5, x + 2c)]` is
Let `θ∈(0, π/2)`. If the system of linear equations,
(1 + cos2θ)x + sin2θy + 4sin3θz = 0
cos2θx + (1 + sin2θ)y + 4sin3θz = 0
cos2θx + sin2θy + (1 + 4sin3θ)z = 0
has a non-trivial solution, then the value of θ is
______.
