I see there is a bit of confusion, so I'll try to explain better.
A matrix $A$ is positive semidefinite (p.sd.) if it is symmetric and all its eigenvalues are $\ge 0$.
A matrix $A$ is positive definite (p.d.) if it is symmetric and all its eigenvalues are $>0$.
This means that every p.d. matrix is also a p.sd. matrix. The set of positive semidefinite matrices contains the set of positive definite matrices, but there are some p.sd. matrices that are not p.d., like the matrix that has all elements equal to zero.
As an analogy, you can think about the real and integer numbers: all integer numbers are real numbers, because the set of real numbers contains the set of integer numbers, but there are real numbers, such as $\sqrt 2$ that are not integer.
The most common example is the identity matrix $I$: all its eigenvalues are $1>0$, so it is a p.d. matrix, but $1\ge 0$ so it is also a p.sd. matrix. This means that $I$ is a p.sd. matrix and it is invertible.
As Algebraic Pavel stated correctly:
Every principal submatrix of a positive (semi)definite matrix is positive (semi)definite.
Here some elaborate statements:
If $A\in\mathbb{R}^{n\times n}$ is positive definite, then all of its principal submatrices $a_{1:m,1:m}$ ($m=1,\dots, n$) are positive definite. If $A$ is positive semi-definite, then all of its principal submatrices $a_{1:m,1:m}$ ($m=1,\dots, n$) are positive semi-definite.
This also works for negative (semi)-definite matrices, by simply multiplying the matrix by -1, i.e.
If $-A\in\mathbb{R}^{n\times n}$ is positive definite, then all of its
principal submatrices $-a_{1:m,1:m}$ ($m=1,\dots, n$) are positive
definite. If $-A$ is positive semidefinite, then all of its principal
submatrices $-a_{1:m,1:m}$ ($m=1,\dots, n$) are positive semidefinite.
For a reference, see Observation 7.1.2 from Matrix Analysis (Horn, Johnson), 2nd edition.
See also Sylvester's criterion, which is a similar statement regarding principle minors.
Best Answer
For a symmetric matrix $B$, it follows from Sylverster's criterion that this is not possible. Positive definiteness requires that $b_{11} > 0$. The second leading principal minor is $-b_{21}b_{21} < 0,$ thus the matrix is also not positive semi-definite.
For non-symmetric matrices $B$, there is no general agreement on what a "positive definite non-symmetric matrix" is.