A positive matrix must be a square matrix
true or false
The correct answer and explanation is:
The statement “A positive matrix must be a square matrix” is false.
A matrix is termed positive if all of its elements are positive numbers. This condition is independent of whether the matrix is square or rectangular. A square matrix is one where the number of rows equals the number of columns, while a rectangular matrix has a different number of rows and columns. A matrix being positive only means that each individual entry (or element) is greater than zero.
For example, a 3×2 matrix: (123456)\begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix}
is a positive matrix since all its entries are positive, even though it is not square (it has 3 rows and 2 columns). Therefore, the matrix does not need to be square to be considered positive.
A square matrix is often important in other contexts, such as in linear algebra for discussing determinants, eigenvalues, and matrix inverses. However, the positivity of a matrix refers only to the individual values of the elements, and not the shape or dimension of the matrix.