For what values of (¢ belongs to R) the following system of equations has (1)no solution (2)a unique solution (3)infinitely many solutions?
(5-¢)x+4y+2z=4
4x+(5-¢)y+2z=4
2x+2y+(2-¢)z=2
2) Consider three planes in R3. List all possibilities of their intersection. Give example of each case(equations,not diagrams)
3) Let A=|£ ¢| be a
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matrix and (b vector) belongs to R2. Consider the system represented by AX=b. Then show that
(a) if A is a zero matrix, then any vector (x vector) belongs to R2 is a solution to the system
1) To determine the values of ¢ for which the system of equations has no solution, a unique solution, or infinitely many solutions, we can use the concept of determinants.
First, let's write the system of equations in matrix form:
⎡ 5 - ¢ 4 2 ⎤ ⎡ x ⎤ ⎡ 4 ⎤
⎢ 4 5 - ¢ 2 ⎥ * ⎢ y ⎥ = ⎢ 4 ⎥
⎣ 2 2 2 - ¢⎦ ⎣ z ⎦ ⎣ 2 ⎦
Now, calculate the determinant of the coefficient matrix A:
|A| = ⎡ 5 - ¢ 4 2 ⎤
⎢ 4 5 - ¢ 2 ⎥
⎣ 2 2 2 - ¢⎦
Expand the determinant using the first row:
|A| = (5 - ¢) [(5 - ¢)(2 - ¢) - 2(2)] - 4 [(4)(2 - ¢) - 2(2)] + 2 [(4)(2) - (5 - ¢)(2)]
Now simplify and solve the resulting equation to find the values of ¢:
(5 - ¢)[(5 - ¢)(2 - ¢) - 4] - 4[8 - 4¢ - 4] + 2[8 - 10 + ¢] = 0
Simplify further:
(5 - ¢)[2¢² - 9¢ + 6] - 4(-4¢) + 2(¢ - 2) = 0
Now, solve this quadratic equation by factoring or using the quadratic formula to find the values of ¢ for which the determinant is zero:
2¢² - 9¢ + 6 - 4¢² + 16¢ + 8 + 2¢ - 4 = 0
-2¢² + 9¢ + 10 = 0
Solve this quadratic equation to find the values of ¢:
(2¢ - 5)(¢ - 2) = 0
2¢ - 5 = 0 -> ¢ = 2.5
¢ - 2 = 0 -> ¢ = 2
So, when ¢ = 2 and ¢ = 2.5, the system of equations will have infinitely many solutions.
To check for a unique solution, substitute the found values of ¢ back into the original equations and solve the system of equations using Gaussian elimination or any other suitable method.
2) There are several possibilities for the intersection of three planes in R3:
- There can be a single point of intersection, forming a unique solution to the system of equations representing the planes.
- The planes can intersect in a line, meaning there are infinitely many solutions to the system of equations.
- The planes can be parallel to each other, resulting in no intersection and no solution.
Here's an example of each case:
(a) Unique solution:
Consider the following system of equations representing three planes:
x + y + z = 3
2x - y + z = 1
3x + 2y - z = 4
Solve the system using any method, such as Gaussian elimination, to find the unique solution (x, y, z).
(b) Infinitely many solutions:
Consider the system of equations representing three planes:
x + y + z = 2
2x - 2y + 2z = 4
3x - 3y + 3z = 6
Solve the system using any method, and you'll find that the equations are consistent and dependent, meaning that there are infinitely many solutions. The planes intersect along a line.
(c) No solution:
Consider the following system of equations representing three planes:
x + y + z = 1
2x + 2y + 2z = 4
3x + 3y + 3z = 3
Upon solving the system, you'll find that the last equation is contradictory to the first two equations. There is no solution, and the planes do not intersect.
3) Let's consider the system represented by AX = b, where A is a matrix and b is a vector in R2.
(a) If A is a zero matrix, then any vector x in R2 is a solution to the system.
A zero matrix is a matrix in which all entries are zero. If A is a zero matrix, then the system represented by AX = b becomes a system of equations where each equation is 0 = b. Since the left-hand side of the equations is always zero, any vector x in R2 could satisfy the equation since multiplying a zero matrix by any vector will yield zero.
For example, if A is the zero matrix and b = [3 2], the system would be:
0x + 0y = 3
0x + 0y = 2
Since both equations are always true (0 = 3 is always false, and 0 = 2 is also always false), any vector x = [x₁ x₂] would be a solution to the system.
Note: In general, a system with a zero matrix as the coefficient matrix will have either no solutions (if b ≠ 0) or infinitely many solutions (if b = 0).