To find the LU decomposition of Matrix A given as:
Explanation of Solution
LU Decomposition of A: The LU Decomposition of A means to decompose a matrix A into a product of lower triangular matrix and upper triangular matrix.
In Decomposition of A the input is lower triangular matrix and output is upper triangular matrix obtain by applying following row operation on A.
Here,
and
And lower triangular matrix is given by
Consider a matrix,
Initially calculate the value of all entries other than 0 and 1 of equation (1).
Apply the row operation to the matrix A to find the upper triangular matrix U
Calculate the value of 
(2)
Calculate the value of 
,
(3)
Calculate the value of 
(4)
Calculate the value of 
(5)
Apply row transformation 
to find upper triangular matrix.
Similarly calculate the value of,
…… (6)
Calculate the value of,
…… (7)
…… (8)
Apply row transformation 
to find upper triangular matrix.
Calculate the value of,
(9)
Calculate the value of,
(10)
Apply row transformation 
to find upper triangular matrix.
The upper triangular matrix is-
The lower triangular matrix is-
Hence, the LU decomposition of A is-
To solve the equation 
by using forward and backward substitution.
Explanation of Solution
Consider an expression-
To solve this, the following steps are followed-
Express 
in the 
decomposition form-
Let for any
• LUP Decomposition
• Forward Substitution
Let 
After solving, the following value of 
is obtained-
• Backward Substitution
Solving for 
To find the inverse of A.
Explanation of Solution
Inverse of a matrix 
is a matrix 
such that 
where 
is an identity matrix.
To solve this, the following steps are followed-
• Forward Substitution
Let 
After solving, the following value of 
is obtained-
Similarly, other columns can also be obtained in the following way-
Inverse of the matrix can be formed by using the five results which comprises the columns of the inverse matrix.
• Backward Substitution
Now find 
from 
Solving for 
To show how, for any 
symmetric positive-definite, tridiagonal matrix 
and any n-vector 
to solve the equation
Explanation of Solution
Inverse of a matrix 
is a matrix 
such that 
where 
is an identity matrix.
To solve this, the following steps are followed-
• Forward Substitution
Let 
After solving, the following value of 
is obtained-
Similarly, other columns can also be obtained in the following way-
Inverse of the matrix can be formed by using the five results which comprises the columns of the inverse matrix.
• Backward Substitution
Now find 
from 
Solving for 
To show that, for any 
non-singular, tridiagonal matrix.
Explanation of Solution
Forward and Backward substitution take the same amount of operations as in the previous case. The only different case is the decomposition step. In this case, the decomposition is done into unit lower triangular and upper triangular matrix, and. The pivoting is done by the swapping rows such that the highest element of a column comes to the diagonal place. The decomposition looks like the following:
Here, 
loses the tri-diagonal property of matrix
but it is still a banded matrix. The essence to solve this is similar to the tri-diagonal case. The time complexity now depends on the width of the band or the distance between the farthest diagonals of the resultant matrix. Matrix becomes wider than the upper triangle of the initial matrix. The final complexity becomes
Hence the time complexity of LUP decomposition 
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