(1 point) which of the following subsets of r3×3 are subspaces of r3×3? (r3×3 denotes the set of all 3×3 matrices with real entries.)a. the symmetric 3×3 matricesb. the 3×3 matrices whose entries are all integersc. the 3×3 matrices with all zeros in the third rowd. the 3×3 matrices in reduced row-echelon forme. the invertible 3×3 matricesf. the diagonal 3×3 matrices

For all the subset proposed, we have to ask ourself:

So, we have:

A. The symmetric 3×3 matrices

Symmetric matrices are equal to their transpose. So, the multiple of a symmetric matrix is still symmetric, and the sum of two symmetric is still symmetric, because

So, this is a subspace as well.

B. The 3×3 matrices whose entries are all integers

This is not a subspace, because we are allowed to use real scalars. So, if A is a matrix whose entries are all integers, we may pick an irrational scalar and we'd have, for example, . The entries of this matrix are not integers, so the subset is not close under scalar multiplication, and thus it's not a subspace

C. The 3×3 matrices with all zeros in the third row

This is a subspace: if you multiply a matrix with a zero-row by a scalar, that row will remain zero. Similarly, if you sum two such matrices, the third row will result in a sum of zeroes.

D. The 3×3 matrices in reduced row-echelon form

This follows the same logic of case C: the zeroes remain where they are, so this is a subspace

E. The invertible 3×3 matrices

This is not a subspace: if A is invertible so is -A, but their sum A-A=0 is the null matrix, which is not invertible

F. The diagonal 3×3 matrices

This is a subspace: if A has nonzero elements only on the diagonal, so does . Similarly, if A and B are diagonal, A+B will have non-zero addends only on the diagonal.

step-by-step explanation:

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question: how many people can she serve?

answer: she can serve 16.66666667 people. so she can only give 16 people a complete serving.

question: does she have enough ice cream for her guests?

answer: no. explanation: she only has enough for 16 people to have a full serving. however, she can give a 17th person a partial serving.