Advertisements
Advertisements
Question
If A = `[[1,1,1],[0,1,3],[1,-2,1]]` , find A-1Hence, solve the system of equations:
x +y + z = 6
y + 3z = 11
and x -2y +z = 0
Solution
A = `[[1,1,1],[0,1,3],[1,-2,1]]`
A11 = 7 , A12 = 3, A13 = -1
A21 = -3 , A22 = 0, A23 = +3
A31 = 2, A32 = -3, A33 = 1
|A| = 1(7) + 3 - 1= 9
`∴ A^(-1) = 1/|A| adj A`
` = 1/9[[7,-3,2],[3,0,-3],[-1,3,1]]`
Verification
AA-1 = I
`= 1/9[[1,1,1],[0,1,3],[1,-2,1]] xx [[7,-3,2],[3,0,-3],[-1,3,1]] `
`= 1/9[[9,0,0],[0,9,0],[0,0,9]]`
=I3
X +Y + Z = 6
0X + Y + 3Z = 11
X -2Y + Z = 0
`[[1,1,1],[0,1,3],[1,-2,1]] [ [X],[Y],[Z]] =[[6],[11],[0]]`
aX =b ⇒ x = A-1 b
`A^(-1) = 1/9 [[7,-3,2],[3,0,-3],[-1,3,1]] `
`∴ [[x],[y],[x]] =A^(-1)b`
`= 1/9 [[7,-3,2],[3,0,-3],[-1,3,1]] [[6],[11],[0]]`
`=1/9 [[42-33],[18],[-6+33]] =1/9 [[9],[18],[27]]`
`=[[1],[2],[3]]`
∴ x =1; y =2; z = 3
APPEARS IN
RELATED QUESTIONS
Evaluate the following determinant:
\[\begin{vmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}a & h & g \\ h & b & f \\ g & f & c\end{vmatrix}\]
Prove the following identities:
\[\begin{vmatrix}x + \lambda & 2x & 2x \\ 2x & x + \lambda & 2x \\ 2x & 2x & x + \lambda\end{vmatrix} = \left( 5x + \lambda \right) \left( \lambda - x \right)^2\]
Prove the following identity:
\[\begin{vmatrix}2y & y - z - x & 2y \\ 2z & 2z & z - x - y \\ x - y - z & 2x & 2x\end{vmatrix} = \left( x + y + z \right)^3\]
Solve the following determinant equation:
Using determinants show that the following points are collinear:
(3, −2), (8, 8) and (5, 2)
Prove that :
Prove that :
Prove that
2x − y = − 2
3x + 4y = 3
9x + 5y = 10
3y − 2x = 8
3x + y + z = 2
2x − 4y + 3z = − 1
4x + y − 3z = − 11
x − y + z = 3
2x + y − z = 2
− x − 2y + 2z = 1
2x + y − 2z = 4
x − 2y + z = − 2
5x − 5y + z = − 2
x − y + 3z = 6
x + 3y − 3z = − 4
5x + 3y + 3z = 10
An automobile company uses three types of steel S1, S2 and S3 for producing three types of cars C1, C2and C3. Steel requirements (in tons) for each type of cars are given below :
| Cars C1 |
C2 | C3 | |
| Steel S1 | 2 | 3 | 4 |
| S2 | 1 | 1 | 2 |
| S3 | 3 | 2 | 1 |
Using Cramer's rule, find the number of cars of each type which can be produced using 29, 13 and 16 tons of steel of three types respectively.
Solve each of the following system of homogeneous linear equations.
3x + y + z = 0
x − 4y + 3z = 0
2x + 5y − 2z = 0
If \[A = \left[ a_{ij} \right]\] is a 3 × 3 diagonal matrix such that a11 = 1, a22 = 2 a33 = 3, then find |A|.
Find the value of the determinant \[\begin{vmatrix}243 & 156 & 300 \\ 81 & 52 & 100 \\ - 3 & 0 & 4\end{vmatrix} .\]
Write the value of the determinant \[\begin{vmatrix}2 & - 3 & 5 \\ 4 & - 6 & 10 \\ 6 & - 9 & 15\end{vmatrix} .\]
If ω is a non-real cube root of unity and n is not a multiple of 3, then \[∆ = \begin{vmatrix}1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1\end{vmatrix}\]
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
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
Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prices at the rate of Rs. x, y and z respectively per person. The first institution decided to award respectively 4, 3 and 2 employees with a total price money of Rs. 37000 and the second institution decided to award respectively 5, 3 and 4 employees with a total price money of Rs. 47000. If all the three prices per person together amount to Rs. 12000 then using matrix method find the value of x, y and z. What values are described in this equations?
`abs ((("b" + "c"^2), "a"^2, "bc"),(("c" + "a"^2), "b"^2, "ca"),(("a" + "b"^2), "c"^2, "ab")) =` ____________.
Choose the correct option:
If a, b, c are in A.P. then the determinant `[(x + 2, x + 3, x + 2a),(x + 3, x + 4, x + 2b),(x + 4, x + 5, x + 2c)]` 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 ______.
Using the matrix method, solve the following system of linear equations:
`2/x + 3/y + 10/z` = 4, `4/x - 6/y + 5/z` = 1, `6/x + 9/y - 20/z` = 2.
