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There is no implementation for substituting a matrix into a symbolic expression, because the operation is not well-defined in general. (For example, what should happen when you substitute a matrix into exp(-1/t)?)

Of course it is well-defined for polynomials. This substitution is implemented, but only for polynomials as members of a polynomial ring (rather than symbolic expressions), so you have to do a conversion:

sage: t2.polynomial(QQ).subs(t=I4)
[16  0  0  0  0]
[ 0 16  0  0  0]
[ 0  0 16  0  0]
[ 0  0  0 16  0]
[ 0  0  0  0 16]

It is easier (in life in general) to avoid symbolic expressions altogether, and to define t as a generator of a polynomial ring (instead of a symbolic variable), so that substitutions into polynomials in t work immediately:

sage: t = polygen(QQ, name='t')
sage: t^2
t^2
sage: (t^2).subs(t=I4)
[16  0  0  0  0]
[ 0 16  0  0  0]
[ 0  0 16  0  0]
[ 0  0  0 16  0]
[ 0  0  0  0 16]