Determine which of the following subsets of R33are subspaces

Determine which of the following subsets of R^3×3are subspaces of R^3×3

YES OR NO ANSWERS FOR THE FOLLOW QUETIONS:

1.) The singular 3×3 matrices

2.) The upper triangular 3×3 matrices

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

4.) The 3×3 matrices with trace 0 (the trace of a matrix is the sum of its diagonal entries)

5.) The invertible 3×3 matrices

6.) The 3×3 matrices in reduced row-echelon form

7.) The 3×3 matrices whose entries are all integers

8.) The symmetric 3×3 matrices

Solution

For a subset to be subspace it must follow the following three conditions:

(i) Should contain zero vector

(ii) Shoud be closed under vector addition

(iii) Should be closed under scalar multiplication

Let us now check all the options for these three conditions

(1) Fails closeness under vector addition. Hence this is a \'NO\'

(2) Satisfies all three conditions. Hence this is a \'Yes\'

(3) Satisfies all three conditions. Hence this is a \'Yes\'

(4) Fails closeness under vector addition. Hence this is a \"NO\'

(5) Fails closeness under vector addition. Hence this is a \'NO\'

(6) Satisfies all the conditions. Hence this is a \'Yes\'

(7) Fails closeness under scalar multiplication. Hence this is a \'NO\'

8) Satisfies all the conditions. Hence this is a \'Yes\'