For each of the following statements, indicate whether the statement is true or

Question

For each of the following statements, indicate whether the statement is true or false. If a triangular matrix has a zero on its main diagonal, then it is necessarily singular. The product of two upper triangular matrices is upper triangular. If a linear system is well-conditioned, then pivoting is unnecessary in Gaussian elimination. Once the LU factorization of a matrix has been computed to solve a linear system, then subsequent linear systems with the same matrix but different right-hand-side vectors can be solved without refactoring the matrix.

Explanation / Answer

1).true

Determinant can be calculated with ease using triangulated matrix.The determinant of triangulated elements is product of diagonal elements. since diagonal elements are zero the matrix is singular.

2).true

Let P and Q be the two upper triangle matrices with order n*n. According to the row- column matrix multiplication we know the (i-j) th rule of PQ

Pi1Q1j+Pi2Q2j+........+PinQnj

to prove that upper triangle matrix is upper triangle i.e i> j furtherly above expression must rate to zero.

3).true

pivoting can introduce errors in the values of gaussian elimination so pivoting is unnecessary for well condiotioned linear system.

4).true

LU factorsation employs pivoting strategy.

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