Is it a Probability Matrix?

In probability theory, a probability matrix is a matrix such that:

• The matrix is a square matrix (same number of rows as columns).
• All entries are probabilities, i.e. numbers between 0 and 1.
• All rows add up to 1.

The following is an example of a probability matrix:

[
  [0.5, 0.5, 0.0],
  [0.2, 0.5, 0.3],
  [0.1, 0.2, 0.7]
]

Note that though all rows add up to 1, there is no restriction on the columns, which may or may not add up to 1.

Write a function that determines if a matrix is a probability matrix or not.

Examples

is_prob_matrix([
  [0.5, 0.5, 0.0],
  [0.2, 0.5, 0.3],
  [0.1, 0.2, 0.7]
]) ➞ True

is_prob_matrix([
  [0.5, 0.5, 0.0],
  [0.2, 0.5, 0.3]
]) ➞ False

# Not a square matrix.

is_prob_matrix([
  [2, -1],
  [-1, 2]
]) ➞ False

# Entries not between 0 and 1.

is_prob_matrix([
  [0, 1],
  [1, 0]
]) ➞ True

is_prob_matrix([
  [0.5, 0.4],
  [0.5, 0.6]
]) ➞ False

# Rows do not add to 1.

Notes

Fun fact: for most probability matrices M (for example, if M has no zero entries), the matrix powers M^n converge (as n increases) to a matrix where all rows are identical.