[10 points] Given matrix A =  2 2 3, −6 −7 8 (a) (5 points). Show that A has no LU decomposition. (b) (5 points). Find the decomposition PA = LU, where P is an elementary permutation matrix.

Both the answers are as in the solution.

Step-by-step explanation:

As the given matrix is not in the readable form, a similar question is found online and the solution of which is attached herewith.

Part a:

Given matrix is : A =

Here,

Then, A is non-singular matrix.

Here, A₁₁= 0.

If we write A as LU with L lower triangular matrix and U upper triangular matrix, then A₁₁=L₁₁U₁₁.

So, As

A₁₁ = 0 gives L₁₁U₁₁= 0 ,

This indicates that either L₁₁= 0 or U₁₁ = 0.

If L₁₁= 0 or U₁₁ = 0, this would made the corresponding matrix singular, which contradicts the condition as  A is non-singular.

Therefore, A has no LU decomposition.

Part b:

By the implementation of the various row operations

interchange R1 and R2

R3+3R1=R3

R3+(1/3)R2 = R3

Therefore, U = .

Here, LP = E₁₂=E₃₁=-3 &E₃₂=-1/3

So now U is given as

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