Advertisements
Advertisements
Question
Show that each one of the following systems of linear equation is inconsistent:
4x − 5y − 2z = 2
5x − 4y + 2z = −2
2x + 2y + 8z = −1
Advertisements
Solution
The given system of equations can be written as follows:
AX = B
Here,
\[ A = \begin{bmatrix}4 & - 5 & - 2 \\ 5 & - 4 & 2 \\ 2 & 2 & 8\end{bmatrix}, X = \begin{bmatrix}x \\ y \\ z\end{bmatrix}\text{ and }B = \begin{bmatrix}2 \\ - 2 \\ - 1\end{bmatrix}\]
\[\left| A \right| = \begin{vmatrix}4 & - 5 & - 2 \\ 5 & - 4 & 2 \\ 2 & 2 & 8\end{vmatrix}\]
\[ = 4\left( - 32 - 4 \right) + 5\left( 40 - 4 \right) - 2(10 + 8)\]
\[ = - 144 + 180 - 36\]
\[ = 0\]
\[ {\text{ Let }C}_{ij} {\text{ be the cofactors of the elements a }}_{ij}\text{ in }A\left[ a_{ij} \right].\text{ Then,}\]
\[ C_{11} = \left( - 1 \right)^{1 + 1} \begin{vmatrix}- 4 & 2 \\ 2 & 8\end{vmatrix} = 28, C_{12} = \left( - 1 \right)^{1 + 2} \begin{vmatrix}5 & 2 \\ 2 & 8\end{vmatrix} = - 36, C_{13} = \left( - 1 \right)^{1 + 3} \begin{vmatrix}5 & - 4 \\ 2 & 2\end{vmatrix} = 18\]
\[ C_{21} = \left( - 1 \right)^{2 + 1} \begin{vmatrix}- 5 & - 2 \\ 2 & 8\end{vmatrix} = 36 , C_{22} = \left( - 1 \right)^{2 + 2} \begin{vmatrix}4 & - 2 \\ 2 & 8\end{vmatrix} = 36 , C_{23} = \left( - 1 \right)^{2 + 3} \begin{vmatrix}4 & - 5 \\ 2 & 2\end{vmatrix} = - 18\]
\[ C_{31} = \left( - 1 \right)^{3 + 1} \begin{vmatrix}- 5 & - 2 \\ - 4 & 2\end{vmatrix} = - 18, C_{32} = \left( - 1 \right)^{3 + 2} \begin{vmatrix}4 & - 2 \\ 5 & 2\end{vmatrix} = - 18, C_{33} = \left( - 1 \right)^{3 + 3} \begin{vmatrix}4 & - 5 \\ 5 & - 4\end{vmatrix} = 9\]
\[adj A = \begin{bmatrix}28 & - 36 & 18 \\ 36 & 36 & - 18 \\ - 18 & - 18 & 9\end{bmatrix}^T \]
\[ = \begin{bmatrix}28 & 36 & - 18 \\ - 36 & 36 & - 18 \\ 18 & - 18 & 9\end{bmatrix}\]
\[\left( adj A \right)B = \begin{bmatrix}28 & 36 & - 18 \\ - 36 & 36 & - 18 \\ 18 & - 18 & 9\end{bmatrix}\begin{bmatrix}2 \\ - 2 \\ - 1\end{bmatrix}\]
\[ = \begin{bmatrix}56 - 72 + 18 \\ - 72 - 72 + 18 \\ 36 + 36 - 9\end{bmatrix}\]
\[ = \begin{bmatrix}2 \\ - 126 \\ 63\end{bmatrix} \neq 0\]
Hence, the given system of equations is consistent.
APPEARS IN
RELATED QUESTIONS
Examine the consistency of the system of equations.
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
\[∆ = \begin{vmatrix}\cos \alpha \cos \beta & \cos \alpha \sin \beta & - \sin \alpha \\ - \sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{vmatrix}\]
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:
\[\begin{vmatrix}1 & 3 & 5 \\ 2 & 6 & 10 \\ 31 & 11 & 38\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1 & a & a^2 - bc \\ 1 & b & b^2 - ac \\ 1 & c & c^2 - ab\end{vmatrix}\]
\[\begin{vmatrix}0 & b^2 a & c^2 a \\ a^2 b & 0 & c^2 b \\ a^2 c & b^2 c & 0\end{vmatrix} = 2 a^3 b^3 c^3\]
Prove the following identity:
\[\begin{vmatrix}2y & y - z - x & 2y \\ 2z & 2z & z - x - y \\ x - y - z & 2x & 2x\end{vmatrix} = \left( x + y + z \right)^3\]
Find the area of the triangle with vertice at the point:
(2, 7), (1, 1) and (10, 8)
If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.
Using determinants, find the equation of the line joining the points
(3, 1) and (9, 3)
Find values of k, if area of triangle is 4 square units whose vertices are
(k, 0), (4, 0), (0, 2)
Find values of k, if area of triangle is 4 square units whose vertices are
(−2, 0), (0, 4), (0, k)
Prove that :
Prove that :
Prove that :
2x − y = 17
3x + 5y = 6
2y − 3z = 0
x + 3y = − 4
3x + 4y = 3
3x + y = 5
− 6x − 2y = 9
Solve each of the following system of homogeneous linear equations.
2x + 3y + 4z = 0
x + y + z = 0
2x − y + 3z = 0
Write the value of the determinant
\[\begin{bmatrix}2 & 3 & 4 \\ 2x & 3x & 4x \\ 5 & 6 & 8\end{bmatrix} .\]
If \[A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 0 \\ - 1 & 0\end{bmatrix}\] , find |AB|.
If the matrix \[\begin{bmatrix}5x & 2 \\ - 10 & 1\end{bmatrix}\] is singular, find the value of x.
If a > 0 and discriminant of ax2 + 2bx + c is negative, then
\[∆ = \begin{vmatrix}a & b & ax + b \\ b & c & bx + c \\ ax + b & bx + c & 0\end{vmatrix} is\]
If \[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}\] , then x =
Solve the following system of equations by matrix method:
2x + y + z = 2
x + 3y − z = 5
3x + y − 2z = 6
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
Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award ₹x each, ₹y each and ₹z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹1,600. School B wants to spend ₹2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.
If `|(2x, 5),(8, x)| = |(6, 5),(8, 3)|`, then find x
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)) =` ____________.
Let A = `[(1,sin α,1),(-sin α,1,sin α),(-1,-sin α,1)]`, where 0 ≤ α ≤ 2π, then:
The number of real values λ, such that the system of linear equations 2x – 3y + 5z = 9, x + 3y – z = –18 and 3x – y + (λ2 – |λ|z) = 16 has no solution, is ______.
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 ab + bc + ca equals ______.
