Advertisements
Advertisements
प्रश्न
3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20.
Advertisements
उत्तर
Given: 3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20
\[D = \begin{vmatrix}3 & - 1 & 2 \\ 2 & - 1 & 1 \\ 3 & 6 & 5\end{vmatrix}\]
\[3\left( - 5 - 6 \right) + 1\left( 10 - 3 \right) + 2\left( 12 + 3 \right) = 4\]
Since D is non-zero, the system of linear equations is consistent and has a unique solution.
\[ D_1 = \begin{vmatrix}6 & - 1 & 2 \\ 2 & - 1 & 1 \\ 20 & 6 & 5\end{vmatrix}\]
\[ = 6\left( - 5 - 6 \right) + 1\left( 10 - 20 \right) + 2\left( 12 + 20 \right)\]
\[ = - 66 - 10 + 64\]
\[ = - 12\]
\[ D_2 = \begin{vmatrix}3 & 6 & 2 \\ 2 & 2 & 1 \\ 3 & 20 & 5\end{vmatrix}\]
\[ = 3\left( 10 - 20 \right) - 6\left( 10 - 3 \right) + 2\left( 40 - 6 \right)\]
\[ = - 30 - 42 + 68\]
\[ = - 4\]
\[ D_3 = \begin{vmatrix}3 & - 1 & 6 \\ 2 & - 1 & 2 \\ 3 & 6 & 20\end{vmatrix}\]
\[ = 3\left( - 20 - 12 \right) + 1\left( 40 - 6 \right) + 6\left( 12 + 3 \right)\]
\[ = - 96 + 34 + 90\]
\[ = 28\]
Now,
\[x = \frac{D_1}{D} = \frac{- 12}{4} = - 3\]
\[y = \frac{D_2}{D} = \frac{- 4}{4} = - 1\]
\[z = \frac{D_3}{D} = \frac{28}{4} = 7\]
\[ \therefore x = - 3, y = - 1\text{ and }z = 7\]
APPEARS IN
संबंधित प्रश्न
If `|[2x,5],[8,x]|=|[6,-2],[7,3]|`, write the value of x.
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 + z = 4
2x + y − 3z = 0
x + y + z = 2
Evaluate the following determinant:
\[\begin{vmatrix}6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\left( 2^x + 2^{- x} \right)^2 & \left( 2^x - 2^{- x} \right)^2 & 1 \\ \left( 3^x + 3^{- x} \right)^2 & \left( 3^x - 3^{- x} \right)^2 & 1 \\ \left( 4^x + 4^{- x} \right)^2 & \left( 4^x - 4^{- x} \right)^2 & 1\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}a & b + c & a^2 \\ b & c + a & b^2 \\ c & a + b & c^2\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}0 & x y^2 & x z^2 \\ x^2 y & 0 & y z^2 \\ x^2 z & z y^2 & 0\end{vmatrix}\]
Prove that:
`[(a, b, c),(a - b, b - c, c - a),(b + c, c + a, a + b)] = a^3 + b^3 + c^3 -3abc`
Prove that
\[\begin{vmatrix}- bc & b^2 + bc & c^2 + bc \\ a^2 + ac & - ac & c^2 + ac \\ a^2 + ab & b^2 + ab & - ab\end{vmatrix} = \left( ab + bc + ca \right)^3\]
Show that
Solve the following determinant equation:
Using determinants, find the equation of the line joining the points
(1, 2) and (3, 6)
Prove that :
Prove that :
2x − y = − 2
3x + 4y = 3
5x + 7y = − 2
4x + 6y = − 3
x − 4y − z = 11
2x − 5y + 2z = 39
− 3x + 2y + z = 1
For what value of x, the following matrix is singular?
If A = [aij] is a 3 × 3 scalar matrix such that a11 = 2, then write the value of |A|.
If I3 denotes identity matrix of order 3 × 3, write the value of its determinant.
If \[\begin{vmatrix}2x + 5 & 3 \\ 5x + 2 & 9\end{vmatrix} = 0\]
If x ∈ N and \[\begin{vmatrix}x + 3 & - 2 \\ - 3x & 2x\end{vmatrix}\] = 8, then find the value of x.
The value of the determinant
If x, y, z are different from zero and \[\begin{vmatrix}1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z\end{vmatrix} = 0\] , then the value of x−1 + y−1 + z−1 is
Solve the following system of equations by matrix method:
3x + 4y + 7z = 14
2x − y + 3z = 4
x + 2y − 3z = 0
Show that each one of the following systems of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5
If \[A = \begin{bmatrix}1 & 2 & 0 \\ - 2 & - 1 & - 2 \\ 0 & - 1 & 1\end{bmatrix}\] , find A−1. Using A−1, solve the system of linear equations x − 2y = 10, 2x − y − z = 8, −2y + z = 7
x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0
The system of linear equations:
x + y + z = 2
2x + y − z = 3
3x + 2y + kz = 4 has a unique solution if
Consider the system of equations:
a1x + b1y + c1z = 0
a2x + b2y + c2z = 0
a3x + b3y + c3z = 0,
if \[\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}\]= 0, then the system has
On her birthday Seema decided to donate some money to children of an orphanage home. If there were 8 children less, everyone would have got ₹ 10 more. However, if there were 16 children more, everyone would have got ₹ 10 less. Using the matrix method, find the number of children and the amount distributed by Seema. What values are reflected by Seema’s decision?
Solve the following system of equations by using inversion method
x + y = 1, y + z = `5/3`, z + x = `4/3`
Using determinants, find the equation of the line joining the points (1, 2) and (3, 6).
If `alpha, beta, gamma` are in A.P., then `abs (("x" - 3, "x" - 4, "x" - alpha),("x" - 2, "x" - 3, "x" - beta),("x" - 1, "x" - 2, "x" - gamma)) =` ____________.
If `|(x + 1, x + 2, x + a),(x + 2, x + 3, x + b),(x + 3, x + 4, x + c)|` = 0, then a, b, care in
Let P = `[(-30, 20, 56),(90, 140, 112),(120, 60, 14)]` and A = `[(2, 7, ω^2),(-1, -ω, 1),(0, -ω, -ω + 1)]` where ω = `(-1 + isqrt(3))/2`, and I3 be the identity matrix of order 3. If the determinant of the matrix (P–1AP – I3)2 is αω2, then the value of α is equal to ______.
