Compute the LU factorization of each of the following matrices. [−21241−2−6−34]

Question

Compute the LU factorization of each of the following matrices.  [−21241−2−6−34]

Arham Warner · Accepted Answer

Step 1  Given: Matrix   A=[−21241−2−6−34]  To find: LU decomposition of the given matrix  Procedure: We are trying to find two matrices L and U such that A = LU  Here, L is a lower triangular matrix whose all diagonal entries are 1 and entries above the main diagonal are 0  Similarly, U is an upper triangular matrix whose all entries below the main diagonal are 0  Hence we start by finding U, by making entries below main diagonal 0 using row operations and then we will find L using opposites of the multiples used in building U  Note that L is of the form:  L=[100101]  Step 2  Computation of U and L  We need to make entry in the first column below -2 as 0   Apply row operation to get second row of U: R2→2R1+R2 and now the multiplier is 2 and so we get -2 below first main diagonal element of L  ⇒A∼[−212032−6−34]L=[100−2101]  Now we want to get 0 in place of -6 in the first column  Apply row operation, R3→3R1+R3 and now the multiplier is -3, hence we get 3 below the entry -2 in L  ⇒A∼[−212032002]L=[100−2103−21]  Call the matrix obtained from A by applying row operations as U  Hence,  ⇒U=[−212032002]L=[100−2103−21]  ⇒[−21241−2−6−34]=[100−2103−21][−212032002]  Step 3  Answer:  [−21241−2−6−34]=[100−2103−21][−212032002]

Compute the LU factorization of each of the following matrices. \begin{bmatrix}-2 & 1 & 2 \\4 & 1 & -2 \\-6 & -3 & 4\end{bmatrix}

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