(a) Solve the following system of equations by LU decomposition without pivoting
(b) Determine the matrix inverse. Check your results by verifying that
(a)
To calculate: The solution of the system of equations given below by LU decomposition without pivoting.
Answer to Problem 3P
Solution:
The solution of the system of equations is
Explanation of Solution
Given:
The system of equations,
Formula used:
(1) The forward substitution equations for L can be expressed as,
(2) The backward substitution equation for U can be expressed as,
Calculation:
Consider the system of equations,
The coefficient
And subtracting the result from equation (2).
Thus, multiply equation (1) by
Now subtract this equation from equation (2),
The coefficient
And subtracting the result from equation (3).
Thus, multiply equation (1) by
Now subtract this equation from equation (3),
Now the set of equations is,
The factors
The coefficient
And subtracting the result from equation (5). Thus, multiply equation (4) by
Now, subtract this equation from equation (5),
The factor
Therefore, the LU decomposition is
Now, to find the solution of the given system:
The forward substitution equations for L can be expressed as,
Solve for
Solve for
Solve for
Thus,
Now, perform backward substitution:
Solve for
Solve for
Solve for
Thus,
(b)
To calculate: The matrix inverse for given system of equations and check the result by verifying that
Answer to Problem 3P
Solution:
The matrix inverse is
Explanation of Solution
Given:
The system of equations,
And the LU decomposition is
Formula used:
(1) The forward substitution equations for L can be expressed as,
(2) The backward substitution equation for U can be expressed as,
Calculation:
Consider the given system of equations:
The matrix [A] is:
The lower and upper triangular matrix after decomposition are given as:
The first column of the inverse matrix can be determined by performing the forward substitution solution with a unit vector (with 1 in the first row) of right-hand-side vector.
The forward substitution equations for L can be expressed as,
Where,
Determine D by substituting L and B as shown below,
Solve for
Solve for
Solve for
Hence, the values obtained are
Solve with forward substitution of
This vector can be used as right-hand side vector of equation,
Solve the above matrix by back substitution, which gives the first column of the inverse matrix as:
Similarly, the second column of the inverse matrix can be determined by performing the forward substitution solution with a unit vector (with 1 in the second row) of right-hand-side vector.
The forward substitution equations for L can be expressed as,
Where,
Determine D by substituting L and B as shown below,
Solve for
Solve for
Solve for
Hence, the values obtained are
Solve with forward substitution of
This vector can be used as right-hand side vector of equation,
Solve the above matrix by back substitution, which gives the second column of the inverse matrix as:
Similarly, the third column of the inverse matrix can be determined by performing the forward substitution solution with a unit vector (with 1 in the third row) of right-hand-side vector.
The forward substitution equations for L can be expressed as,
Where,
Determine D by substituting L and B as shown below,
Solve for
Solve for
Solve for
Hence, the values obtained are
Solve with forward substitution of
This vector can be used as right-hand side vector of equation,
Solve the above matrix by back substitution, which gives the third column of the inverse matrix as:
Thus, the inverse matrix is:
Now, check the result obtained.
Hence, verified.
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