Question

What is the dimension of the space of all lower triangular matrices?

A) n x n
B) (n² - n)/2
C) n x (n+1)/2
D) n x (n-1)/2

Answer 1

The dimension of the space of all lower triangular matrices of size n x n is (n(n + 1))/2, corresponding to choice C, n x (n+1)/2.

Step-by-step explanation:

The question is asking what the dimension of the space of all lower triangular matrices of size n x n is. In a lower triangular matrix, the entries above the diagonal are all zero and only the entries on the diagonal and below are potentially non-zero. For each row, the number of non-zero entries in a lower triangular matrix can range from 1 in the first row, 2 in the second row, up to n in the nth row.

To find the total number of potentially non-zero entries, we can sum these up, which forms an arithmetic series: 1 + 2 + ... + n. The sum of this series is (n(n + 1))/2. Therefore, the dimension of the space of lower triangular n x n matrices is given by the formula (n(n + 1))/2, which corresponds to option C, n x (n+1)/2.