Advertisements
Advertisements
Question
Solve the following systems of linear equations by Cramer’s rule:
3x + 3y – z = 11, 2x – y + 2z = 9, 4x + 3y + 2z = 25
Solution
Δ = `|(3, 3, -1),(2, -1, 2),(4, 3, 2)|`
= 3(– 2 – 6) – 3(4 – 8) –1(6 + 4)
= 3(– 8) – 3(– 4) – 1(10)
= – 24 + 12 – 10
= – 22 ≠ 0
Δx = `|(11, 3, -1),(9, -1, 2),(25, 3, 2)|`
= 11 (– 2 – 6) – 3(18 – 50) – 1(27 + 25)
= 11(– 8) – 3(32) – 1(52)
= – 88 + 96 – 52
= – 44
Δy = `|(3, 11, -1),(2, 9, 2),(4, 25, 2)|`
= 3(18 – 50) – 11(4 – 8) – 1(50 – 36)
= 3(32) – 11(4) – 1(14)
= – 96 + 44 – 14
= – 66
Δx = `|(3, 3, 11),(2, -1, 9),(4, 3, 25)|`
= 3(– 25 – 27) – 3(50 – 36) + 11(6 + 4)
= 3(– 52) – 3(14) + 11(10)
= – 156 – 42 + 110
= – 88
By Cramer's rule x = `Delta_x/Delta = (-44)/(-22)` = 2
y = `Delta_y/Delta = (-66)/(-22)` = 3
z = `Delta_z/Delta = (-88)/(-22)` = 4
∴ x = 2, y = 3, z = 4.
APPEARS IN
RELATED QUESTIONS
Four men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work by using matrix inversion method
The prices of three commodities A, B and C are ₹ x, y and z per units respectively. A person P purchases 4 units of B and sells two units of A and 5 units of C. Person Q purchases 2 units of C and sells 3 units of A and one unit of B . Person R purchases one unit of A and sells 3 unit of B and one unit of C. In the process, P, Q and R earn ₹ 15,000, ₹ 1,000 and ₹ 4,000 respectively. Find the prices per unit of A, B and C. (Use matrix inversion method to solve the problem.)
Solve the following systems of linear equations by Cramer’s rule:
5x – 2y + 16 = 0, x + 3y – 7 = 0
Solve the following systems of linear equations by Cramer’s rule:
`3/x - 4/y - 2/z - 1` = 0, `1/x + 2/y + 1/z - 2` = 0, `2/x - 5/y - 4/z + 1` = 0
In a competitive examination, one mark is awarded for every correct answer while `1/4` mark is deducted for every wrong answer. A student answered 100 questions and got 80 marks. How many questions did he answer correctly? (Use Cramer’s rule to solve the problem).
A fish tank can be filled in 10 minutes using both pumps A and B simultaneously. However, pump B can pump water in or out at the same rate. If pump B is inadvertently run in reverse, then the tank will be filled in 30 minutes. How long would it take each pump to fill the tank by itself? (Use Cramer’s rule to solve the problem)
A family of 3 people went out for dinner in a restaurant. The cost of two dosai, three idlies and two vadais is ₹ 150. The cost of the two dosai, two idlies and four vadais is ₹ 200. The cost of five dosai, four idlies and two vadais is ₹ 250. The family has ₹ 350 in hand and they ate 3 dosai and six idlies and six vadais. Will they be able to manage to pay the bill within the amount they had?
An amount of ₹ 65,000 is invested in three bonds at the rates of 6%, 8% and 9% per annum respectively. The total annual income is ₹ 4,800. The income from the third bond is ₹ 600 more than that from the second bond. Determine the price of each bond. (Use Gaussian elimination method.)
A boy is walking along the path y = ax2 + bx + c through the points (– 6, 8), (– 2, – 12), and (3, 8). He wants to meet his friend at P(7, 60). Will he meet his friend? (Use Gaussian elimination method.)
Choose the correct alternative:
If A = `[(1, tan theta/2),(- tan theta/2, 1)]` and AB = I2, then B =
Choose the correct alternative:
If A = `[(costheta, sintheta),(-sintheta, costheta)]` and A(adj A) = `[("k", 0),(0, "k")]`, then k =
Choose the correct alternative:
If A = `[(2, 3),(5, -2)]` be such that λA–1 = A, then λ is
Choose the correct alternative:
If adj A = `[(2, 3),(4, 1)]` and adj B = `[(1, -2),(-3, 1)]` then adj (AB) is
Choose the correct alternative:
If ρ(A) ρ([A|B]), then the system AX = B of linear equations is
Choose the correct alternative:
If 0 ≤ θ ≤ π and the system of equations x + (sin θ)y – (cos θ)z = 0, (cos θ) x – y + z = 0, (sin θ) x + y + z = 0 has a non-trivial solution then θ is
Choose the correct alternative:
The augmented matrix of a system of linear equations is `[(1, 2, 7, 3),(0, 1, 4, 6),(0, 0, lambda - 7, mu + 7)]`. This system has infinitely many solutions if
Choose the correct alternative:
Let A = `[(2, -1, 1),(-1, 2, -1),(1, -1, 2)]` and 4B = `[(3, 1, -1),(1, 3, x),(-1, 1, 3)]`. If B is the inverse of A, then the value of x is
