45 POINTS FOR THIS ONE, WILL MARK BRAINLIEST FOR FULL CORRECT ANSWER From the following subsets of M(mxn), say whether they are subspaces.
(i) All matrices with a₁₁ = 0.
(ii) All matrices with a₁₁ = 3.
(iii) All matrices whose first column matches the last column.
(iv) All symmetric matrices (i. e., all A matrices with A^t = A).
(v) All antisymmetric matrices (i. e. all A matrices with A^t = A).
(vi) All diagonal arrays.
(vii) All upper triangular matrices.
(viii) All invertible matrices.
(ix) All non-invertible matrices.
(x) All matrices with detA = 1.

v) All antisymmetric matrices (i.e. all A matrices with A^t = A).

(vi) All diagonal arrays.

(vii) All upper triangular matrices.

Step-by-step explanation:

All diagonal arrays.

(vii) All upper triangular matrices.

(viii) All invertible matrices.

(ix) All non-invertible matrices.OR THIS ONE, WILL MARK BRAINLIEST FOR FULL CORRECT ANSWER

From the following subsets of M(mxn), say whether they are subspaces.

(i) All matrices with a₁₁ = 0.

(ii) All matrices with a₁₁ = 3.

(iii) All matrices whose first column matches the last column.

(iv) All symmetric matrices (i.e., all A matrices with A^t = A).

(v) All antisymmetric matrices (i.e. all A matrices with A^t = A).

(vi) AllAll matrices with a₁₁ = 0.

(ii) All matrices with a₁₁ = 3.

(iii) All matrices whose first column matches the last column.

(iv) All symmetric matrices (i.e., all A matrices with A^t = A).

(v) All antisymmetric matrices (i.e. all A matrices with A^t = A).

(vi) All diagonal arrays.

(vii) All upper triangular matrices.

(viii) All invertible matrices.

(ix) All non-invertible matrices.

(x) All matrices with detA = 1.

(x) All matrices with detA = 1.

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.