Advertisements
Advertisements
प्रश्न
Examine the consistency of the system of equations.
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
Advertisements
उत्तर
Let, A = `[(3,-1,-2),(0,2,-1),(3,-5,0)]`, X = `[(x),(y),(z)]`, B = `[(2),(-1),(3)]`
|A| `= |(3,-1,-2),(0,2,-1),(3,-5,0)|`
= 3[2 × 0 + 5 × (−1)] + 1(0 + 3) − 2(0 − 6)
= −15 + 3 + 12
= 0
Cofactors of the elements of |A|:
A11 = `|(2,-1),(-5,0)|`
= 0 − 5
= −5
A12 = `-|(0,-1),(3,0)|`
= −(0 + 3)
= −3
A13 = `|(0,2),(3,-5)|`
= 0 − 6
= −6
A21 = `-|(-1,-2),(-5,0)|`
= −(0 − 10)
= 10
A22 = `|(3,-2),(3,0)|`
= 0 + 6
= 6
A23 = `-|(3,-1),(3,-5)|`
= −(−15 + 3)
= 12
A31 = `|(-1,-2),(2,-1)|`
= 1 + 4
= 5
A32 = `-|(3,-2),(0,-1)|`
= −(−3 + 0)
= 3
A33 = `|(3,-1),(0,2)|`
= 6 + 0
= 6
∴ |A| = `[(-5,-3,-6),(10,6,12),(5,3,6)]`
∴ (adj A) = `[(-5,10,5),(-3,6,3),(-6,12,6)]`
(adj A)B = ` [(-5,10,5),(-3,6,3),(-6,12,6)] [(2),(-1),(3)]`
= `[(-10-10 + 15),(-6 - 6 + 9),(-12 - 12 + 18)]`
= `[(-5),(-3),(-6)] ≠ 0`
∴ |A| = 0 and (adj A)B ≠ 0
Hence, the system of equations is inconsistent.
APPEARS IN
संबंधित प्रश्न
If `|[x+1,x-1],[x-3,x+2]|=|[4,-1],[1,3]|`, then write the value of x.
If \[A = \begin{bmatrix}2 & 5 \\ 2 & 1\end{bmatrix} \text{ and } B = \begin{bmatrix}4 & - 3 \\ 2 & 5\end{bmatrix}\] , verify that |AB| = |A| |B|.
Find the value of x, if
\[\begin{vmatrix}2 & 4 \\ 5 & 1\end{vmatrix} = \begin{vmatrix}2x & 4 \\ 6 & x\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\sqrt{23} + \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{46} & 5 & \sqrt{10} \\ 3 + \sqrt{115} & \sqrt{15} & 5\end{vmatrix}\]
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 the following identity:
\[\begin{vmatrix}2y & y - z - x & 2y \\ 2z & 2z & z - x - y \\ x - y - z & 2x & 2x\end{vmatrix} = \left( x + y + z \right)^3\]
If \[a, b\] and c are all non-zero and
If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.
If the points (x, −2), (5, 2), (8, 8) are collinear, find x using determinants.
Using determinants, find the equation of the line joining the points
(3, 1) and (9, 3)
Prove that :
Prove that :
Solve each of the following system of homogeneous linear equations.
2x + 3y + 4z = 0
x + y + z = 0
2x − y + 3z = 0
For what value of x, the following matrix is singular?
If w is an imaginary cube root of unity, find the value of \[\begin{vmatrix}1 & w & w^2 \\ w & w^2 & 1 \\ w^2 & 1 & w\end{vmatrix}\]
If the matrix \[\begin{bmatrix}5x & 2 \\ - 10 & 1\end{bmatrix}\] is singular, find the value of x.
Write the value of the determinant \[\begin{vmatrix}2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x\end{vmatrix}\]
If \[A = \begin{bmatrix}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{bmatrix}\]. Write the cofactor of the element a32.
If x ∈ N and \[\begin{vmatrix}x + 3 & - 2 \\ - 3x & 2x\end{vmatrix}\] = 8, then find 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
If ω is a non-real cube root of unity and n is not a multiple of 3, then \[∆ = \begin{vmatrix}1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1\end{vmatrix}\]
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:
5x + 7y + 2 = 0
4x + 6y + 3 = 0
Solve the following system of equations by matrix method:
2x + 6y = 2
3x − z = −8
2x − y + z = −3
Show that the following systems of linear equations is consistent and also find their solutions:
6x + 4y = 2
9x + 6y = 3
Given \[A = \begin{bmatrix}2 & 2 & - 4 \\ - 4 & 2 & - 4 \\ 2 & - 1 & 5\end{bmatrix}, B = \begin{bmatrix}1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{bmatrix}\] , find BA and use this to solve the system of equations y + 2z = 7, x − y = 3, 2x + 3y + 4z = 17
2x + 3y − z = 0
x − y − 2z = 0
3x + y + 3z = 0
System of equations x + y = 2, 2x + 2y = 3 has ______
If `|(2x, 5),(8, x)| = |(6, -2),(7, 3)|`, then value of x is ______.
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 the system of equations x + ky - z = 0, 3x - ky - z = 0 & x - 3y + z = 0 has non-zero solution, then k is equal to ____________.
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?
The system of linear equations
3x – 2y – kz = 10
2x – 4y – 2z = 6
x + 2y – z = 5m
is inconsistent if ______.
