Advertisements
Advertisements
प्रश्न
3x + y + z = 2
2x − 4y + 3z = − 1
4x + y − 3z = − 11
Advertisements
उत्तर
Given: 3x + y + z = 2
2x − 4y + 3z = − 1
4x + y − 3z = − 11
\[D = \begin{vmatrix}3 & 1 & 1 \\ 2 & - 4 & 3 \\ 4 & 1 & - 3\end{vmatrix}\]
\[ = 3\left( 12 - 3 \right) - 2\left( - 3 - 1 \right) + 4\left( 3 + 4 \right)\]
\[ = 27 + 8 + 28\]
\[ = 63\]
\[ D_1 = \begin{vmatrix}2 & 1 & 1 \\ - 1 & - 4 & 3 \\ - 11 & 1 & - 3\end{vmatrix}\]
\[ = 2\left( 12 - 3 \right) + 1\left( - 3 - 1 \right) - 11\left( 3 + 4 \right)\]
\[ = 18 - 4 - 77\]
\[ = - 63\]
\[ D_2 = \begin{vmatrix}3 & 2 & 1 \\ 2 & - 1 & 3 \\ 4 & - 11 & - 3\end{vmatrix}\]
\[ = 3\left( 3 + 33 \right) - 2\left( - 6 + 11 \right) + 4\left( 6 + 1 \right)\]
\[ = 108 - 10 + 28\]
\[ = 126\]
\[ D_3 = \begin{vmatrix}3 & 1 & 2 \\ 2 & - 4 & - 1 \\ 4 & 1 & - 11\end{vmatrix}\]
\[ = 3\left( 44 + 1 \right) - 2\left( - 11 - 2 \right) + 4\left( - 1 + 8 \right)\]
\[ = 135 + 26 + 28\]
\[ = 189\]
\[Now, \]
\[x = \frac{D_1}{D} = \frac{- 63}{63} = - 1\]
\[y = \frac{D_2}{D} = \frac{126}{63} = 2\]
\[z = \frac{D_3}{D} = \frac{189}{63} = 3\]
\[ \therefore x = - 1, y = 2\text{ and }z = 3\]
APPEARS IN
संबंधित प्रश्न
Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to ______.
Evaluate the following determinant:
\[\begin{vmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{vmatrix}\]
Find the value of x, if
\[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & 5 \\ 8 & 3\end{vmatrix}\]
For what value of x the matrix A is singular?
\[A = \begin{bmatrix}x - 1 & 1 & 1 \\ 1 & x - 1 & 1 \\ 1 & 1 & x - 1\end{bmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}a & h & g \\ h & b & f \\ g & f & c\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}\]
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 a, b, c are real numbers such that
\[\begin{vmatrix}b + c & c + a & a + b \\ c + a & a + b & b + c \\ a + b & b + c & c + a\end{vmatrix} = 0\] , then show that either
\[a + b + c = 0 \text{ or, } a = b = c\]
Using determinants show that the following points are collinear:
(5, 5), (−5, 1) and (10, 7)
Using determinants show that the following points are collinear:
(1, −1), (2, 1) and (4, 5)
Using determinants show that the following points are collinear:
(3, −2), (8, 8) and (5, 2)
Using determinants, find the area of the triangle with vertices (−3, 5), (3, −6), (7, 2).
Using determinants, find the equation of the line joining the points
(1, 2) and (3, 6)
Prove that :
Prove that :
Prove that :
3x + y = 19
3x − y = 23
2x − y = − 2
3x + 4y = 3
5x + 7y = − 2
4x + 6y = − 3
Given: x + 2y = 1
3x + y = 4
5x − 7y + z = 11
6x − 8y − z = 15
3x + 2y − 6z = 7
Evaluate \[\begin{vmatrix}4785 & 4787 \\ 4789 & 4791\end{vmatrix}\]
If \[A = \left[ a_{ij} \right]\] is a 3 × 3 diagonal matrix such that a11 = 1, a22 = 2 a33 = 3, then find |A|.
For what value of x is the matrix \[\begin{bmatrix}6 - x & 4 \\ 3 - x & 1\end{bmatrix}\] 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
The value of \[\begin{vmatrix}5^2 & 5^3 & 5^4 \\ 5^3 & 5^4 & 5^5 \\ 5^4 & 5^5 & 5^6\end{vmatrix}\]
The number of distinct real roots of \[\begin{vmatrix}cosec x & \sec x & \sec x \\ \sec x & cosec x & \sec x \\ \sec x & \sec x & cosec x\end{vmatrix} = 0\] lies in the interval
\[- \frac{\pi}{4} \leq x \leq \frac{\pi}{4}\]
Solve the following system of equations by matrix method:
3x + y = 19
3x − y = 23
Solve the following equations by using inversion method.
x + y + z = −1, x − y + z = 2 and x + y − z = 3
If `|(2x, 5),(8, x)| = |(6, -2),(7, 3)|`, then value of x is ______.
Let A = `[(1,sin α,1),(-sin α,1,sin α),(-1,-sin α,1)]`, where 0 ≤ α ≤ 2π, then:
For what value of p, is the system of equations:
p3x + (p + 1)3y = (p + 2)3
px + (p + 1)y = p + 2
x + y = 1
consistent?
If `|(x + a, beta, y),(a, x + beta, y),(a, beta, x + y)|` = 0, then 'x' is equal to
If the system of linear equations
2x + y – z = 7
x – 3y + 2z = 1
x + 4y + δz = k, where δ, k ∈ R has infinitely many solutions, then δ + k is equal to ______.
The system of linear equations
3x – 2y – kz = 10
2x – 4y – 2z = 6
x + 2y – z = 5m
is inconsistent if ______.
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
______.
The greatest value of c ε R for which the system of linear equations, x – cy – cz = 0, cx – y + cz = 0, cx + cy – z = 0 has a non-trivial solution, is ______.
