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Consider a system of linear equations PX = Q where P ∈ R^(3×3) and Q ∈ R^(3×1). Suppose P has an LU decomposition, P = LU, where L = [ [1, 0, 0], [l21, 1, 0], [l31, l32, 1] ] and U = [ [u11, u12, u13], [0, u22, u23], [0, 0, u33] ]. Which of the following statement(s) is/are TRUE?

1. The system PX = Q can be solved by first solving LY = Q and then UX = Y.
2. If P is invertible, then both L and U are invertible.
3. If P is singular, then at least one of the diagonal elements of U is zero.
4. If P is symmetric, then both L and U are symmetric.

Correct answer: The system PX = Q can be solved by first solving LY = Q and then UX = Y.

Solution

This statement is correct because the LU decomposition allows us to break down the original system into two simpler systems: first, we solve for Y using the lower triangular matrix L, and then we use the upper triangular matrix U to find the solution X from Y.

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