Advertisements
Advertisements
Question
Find the inverse of the following matrix, using elementary transformations:
`A= [[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]]`
Advertisements
Solution
`A= [[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]]`
AA-1 =I
`[[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]] A^(-1) = [[1 , 0 , 0 ],[0 , 1 , 0],[0 , 0 ,1]]`
R2 → R2 - R1
R3 → R3 - R1
`[[2 , 3 , 1 ],[0 , 1 , 0],[1, 4 ,1]] A^(-1) = [[1 , 0 , 0 ],[-1 , 1 , 0],[-1 , 0 ,1]]`
R1 ↔ R3
`[[1 , 4 , 1 ],[0 , 1 , 0],[2 , 3 ,1]] A^(-1) = [[-1 , 0 , 1 ],[-1 , 1 , 0],[1 , 0 ,0]]`
R3 → R3 - 2R1
`[[1 , 4 , 1 ],[0 , 1 , 0],[0 , -5 ,-1]] A^(-1) = [[-1 , 0 , 1 ],[-1 , 1 , 0],[3 , 0 ,-2]]`
R1 → R1 - 4R2
R3 → R3 - 5R2
`[[1 , 0 , 1 ],[0 , 1 , 0],[0 , 0 ,-1]] A^(-1) = [[3 , -4 , 1 ],[-1 , 1 , 0],[-2 , 5 ,-2]]`
`R_1 ->R_1 +R_3`
`[[1 , 0 , 0 ],[0 , 1 , 0],[0 , 0 ,-1]] A^(-1) = [[1 , 1 , -1 ],[-1 , 1 , 0],[-2 , 5 ,-2]]`
`R_3 -> -R_3`
`[[1 , 0 , 0 ],[0 , 1 , 0],[0 , 0 ,1]] A^(-1) = [[1 , 1 , -1 ],[-1 , 1 , 0],[2 , -5 ,2]]`
`I .A^(1) = [[1 , 1 , -1 ],[-1 , 1 , 0],[2 , -5 ,2]]`
` ⇒A^(1) = [[1 , 1 , -1 ],[-1 , 1 , 0],[2 , -5 ,2]]`
RELATED QUESTIONS
Find the value of a if `[[a-b,2a+c],[2a-b,3c+d]]=[[-1,5],[0,13]]`
The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs. 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs. 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs. 70. Find the cost of each item per kg by matrix method.
Find the value of x, if
\[\begin{vmatrix}2 & 4 \\ 5 & 1\end{vmatrix} = \begin{vmatrix}2x & 4 \\ 6 & x\end{vmatrix}\]
Find the value of x, if
\[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & 5 \\ 8 & 3\end{vmatrix}\]
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:
\[\begin{vmatrix}1/a & a^2 & bc \\ 1/b & b^2 & ac \\ 1/c & c^2 & ab\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}x + \lambda & x & x \\ x & x + \lambda & x \\ x & x & x + \lambda\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]
Show that
If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.
3x + ay = 4
2x + ay = 2, a ≠ 0
If A and B are non-singular matrices of the same order, write whether AB is singular or non-singular.
For what value of x is the matrix \[\begin{bmatrix}6 - x & 4 \\ 3 - x & 1\end{bmatrix}\] singular?
If x ∈ N and \[\begin{vmatrix}x + 3 & - 2 \\ - 3x & 2x\end{vmatrix}\] = 8, then find the value of x.
If \[\begin{vmatrix}x & \sin \theta & \cos \theta \\ - \sin \theta & - x & 1 \\ \cos \theta & 1 & x\end{vmatrix} = 8\] , write the value of x.
If \[\begin{vmatrix}a & p & x \\ b & q & y \\ c & r & z\end{vmatrix} = 16\] , then the value of \[\begin{vmatrix}p + x & a + x & a + p \\ q + y & b + y & b + q \\ r + z & c + z & c + r\end{vmatrix}\] is
Solve the following system of equations by matrix method:
3x + y = 19
3x − y = 23
Solve the following system of equations by matrix method:
3x + y = 7
5x + 3y = 12
Solve the following system of equations by matrix method:
3x + 4y + 2z = 8
2y − 3z = 3
x − 2y + 6z = −2
Solve the following system of equations by matrix method:
x − y + z = 2
2x − y = 0
2y − z = 1
Show that the following systems of linear equations is consistent and also find their solutions:
5x + 3y + 7z = 4
3x + 26y + 2z = 9
7x + 2y + 10z = 5
A company produces three products every day. Their production on a certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product. Determine the production level of each product using matrix method.
Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P 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 ₹2,200. School Q wants to spend ₹3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each values is ₹1,200, using matrices, find the award money for each value.
Apart from these three values, suggest one more value which should be considered for award.
3x − y + 2z = 0
4x + 3y + 3z = 0
5x + 7y + 4z = 0
If `|(2x, 5),(8, x)| = |(6, -2),(7, 3)|`, then value of x is ______.
The system of simultaneous linear equations kx + 2y – z = 1, (k – 1)y – 2z = 2 and (k + 2)z = 3 have a unique solution if k equals:
The number of real value of 'x satisfying `|(x, 3x + 2, 2x - 1),(2x - 1, 4x, 3x + 1),(7x - 2, 17x + 6, 12x - 1)|` = 0 is
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 ______.
If the system of linear equations x + 2ay + az = 0; x + 3by + bz = 0; x + 4cy + cz = 0 has a non-zero solution, then a, b, c ______.
