Question

To prove statements about lower triangular matrices, which of the following holds true?

A) The product of two lower triangular matrices is lower triangular.
B) The inverse of a non-singular lower triangular matrix is lower triangular.
C) Both A and B are correct.
D) Neither A nor B is correct.

Answer 1

The product of two lower triangular matrices is lower triangular.

Step-by-step explanation:

To prove statements about lower triangular matrices, we need to consider the properties of lower triangular matrices.

A) The product of two lower triangular matrices is lower triangular. This statement is true. When you multiply two lower triangular matrices, the resulting matrix will also be lower triangular.

B) The inverse of a non-singular lower triangular matrix is lower triangular. This statement is false. The inverse of a lower triangular matrix may not be lower triangular.

Therefore, the correct answer is Option A) The product of two lower triangular matrices is lower triangular.

Answer 2

The product of two lower triangular matrices is itself lower triangular, and the inverse of a non-singular lower triangular matrix is also lower triangular. Hence, both statements about lower triangular matrices are correct.

Step-by-step explanation:

To address the question regarding lower triangular matrices, let's consider the following statements: (A) The product of two lower triangular matrices is lower triangular, and (B) The inverse of a non-singular lower triangular matrix is lower triangular. For statement A, when we multiply two lower triangular matrices, the result is indeed a lower triangular matrix. This is because when performing matrix multiplication, each element in the product matrix is derived from a dot product that only involves the lower triangular parts of the matrices, which does not introduce any non-zero elements above the diagonal. As for statement B, the inverse of a non-singular lower triangular matrix is lower triangular as well. When inverting a lower triangular matrix, the operation preserves the triangular structure since basic row operations used to find the inverse do not introduce non-zero elements above the diagonal if properly applied. Therefore, the correct answer is C) Both A and B are correct.