Advertisements
Advertisements
Question
3x + y − 2z = 0
x + y + z = 0
x − 2y + z = 0
Advertisements
Solution
The given system of homogeneous equations can be written in matrix form as follows:
\[\begin{bmatrix}3 & 1 & - 2 \\ 1 & 1 & 1 \\ 1 & - 2 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}\]
\[\text{ or, }AX = O\]
\[\text{ where, }A = \begin{bmatrix}3 & 1 & - 2 \\ 1 & 1 & 1 \\ 1 & - 2 & 1\end{bmatrix}, X = \begin{bmatrix}x \\ y \\ z\end{bmatrix}\text{ and }O = \begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}\]
Now,
\[\left| A \right| = \begin{vmatrix}3 & 1 & - 2 \\ 1 & 1 & 1 \\ 1 & - 2 & 1\end{vmatrix}\]
\[ = 3\left( 1 + 2 \right) - 1\left( 1 - 1 \right) - 2\left( - 2 - 1 \right)\]
\[ = 9 - 0 + 6\]
\[ = 15 \neq 0\]
So, the given system has only trivial solution, which is given below:
\[x=y=z=0\]
APPEARS IN
RELATED QUESTIONS
Examine the consistency of the system of equations.
2x − y = 5
x + y = 4
Solve the system of linear equations using the matrix method.
2x – y = –2
3x + 4y = 3
Solve the system of linear equations using the matrix method.
5x + 2y = 3
3x + 2y = 5
Find the value of x, if
\[\begin{vmatrix}x + 1 & x - 1 \\ x - 3 & x + 2\end{vmatrix} = \begin{vmatrix}4 & - 1 \\ 1 & 3\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}a & b + c & a^2 \\ b & c + a & b^2 \\ c & a + b & c^2\end{vmatrix}\]
\[If ∆ = \begin{vmatrix}1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2\end{vmatrix}, ∆_1 = \begin{vmatrix}1 & 1 & 1 \\ yz & zx & xy \\ x & y & z\end{vmatrix},\text{ then prove that }∆ + ∆_1 = 0 .\]
\[\begin{vmatrix}b^2 + c^2 & ab & ac \\ ba & c^2 + a^2 & bc \\ ca & cb & a^2 + b^2\end{vmatrix} = 4 a^2 b^2 c^2\]
Prove the following identity:
`|(a^3,2,a),(b^3,2,b),(c^3,2,c)| = 2(a-b) (b-c) (c-a) (a+b+c)`
Show that x = 2 is a root of the equation
Solve the following determinant equation:
Find the area of the triangle with vertice at the point:
(2, 7), (1, 1) and (10, 8)
Using determinants, find the equation of the line joining the points
(1, 2) and (3, 6)
Prove that :
x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0
x − y + 3z = 6
x + 3y − 3z = − 4
5x + 3y + 3z = 10
Solve each of the following system of homogeneous linear equations.
3x + y + z = 0
x − 4y + 3z = 0
2x + 5y − 2z = 0
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.
For what value of x, the following matrix is singular?
If A and B are non-singular matrices of the same order, write whether AB is singular or non-singular.
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
Let \[\begin{vmatrix}x^2 + 3x & x - 1 & x + 3 \\ x + 1 & - 2x & x - 4 \\ x - 3 & x + 4 & 3x\end{vmatrix} = a x^4 + b x^3 + c x^2 + dx + e\]
be an identity in x, where a, b, c, d, e are independent of x. Then the value of e is
If ω is a non-real cube root of unity and n is not a multiple of 3, then \[∆ = \begin{vmatrix}1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1\end{vmatrix}\]
Let \[A = \begin{bmatrix}1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1\end{bmatrix},\text{ where 0 }\leq \theta \leq 2\pi . \text{ Then,}\]
If \[x, y \in \mathbb{R}\], then the determinant
Solve the following system of equations by matrix method:
6x − 12y + 25z = 4
4x + 15y − 20z = 3
2x + 18y + 15z = 10
Show that the following systems of linear equations is consistent and also find their solutions:
x + y + z = 6
x + 2y + 3z = 14
x + 4y + 7z = 30
A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs 6,000. Three times the award money for Hard work added to that given for honesty amounts to Rs 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.
2x − y + z = 0
3x + 2y − z = 0
x + 4y + 3z = 0
Let a, b, c be positive real numbers. The following system of equations in x, y and z
(a) no solution
(b) unique solution
(c) infinitely many solutions
(d) finitely many solutions
The existence of the unique solution of the system of equations:
x + y + z = λ
5x − y + µz = 10
2x + 3y − z = 6
depends on
If A = `[(1, -1, 2),(3, 0, -2),(1, 0, 3)]`, verify that A(adj A) = (adj A)A
`abs ((("b" + "c"^2), "a"^2, "bc"),(("c" + "a"^2), "b"^2, "ca"),(("a" + "b"^2), "c"^2, "ab")) =` ____________.
The value of λ, such that the following system of equations has no solution, is
`2x - y - 2z = - 5`
`x - 2y + z = 2`
`x + y + lambdaz = 3`
If c < 1 and the system of equations x + y – 1 = 0, 2x – y – c = 0 and – bx+ 3by – c = 0 is consistent, then the possible real values of b are
If the following equations
x + y – 3 = 0
(1 + λ)x + (2 + λ)y – 8 = 0
x – (1 + λ)y + (2 + λ) = 0
are consistent then the value of λ can be ______.
