Which of the following is a subspace of M2x2 (the vector space of 2x2 matrices. and explain why or why not: 1) Set of 2x2 matrices A such that det(A)=1 2) set of 2x2 matrices B such that B[1 -1]^t=0 vector To check if something is a subspace I must satisfy 3 conditions (applied for matrix A): 1) 0 matrix is A 2) If U and V are in A then U+V is in A 3) if V is in A then cV is in A for some scalar c. The above is analogous for matrix B. For 1) Set of 2x2 matrices A such that det(A)=1 The 0 matrix is not in this set because the determinant is 0 which ≠1, thus the set of 2x2 matrices A is not a subspace. Is this correct? For 2) set of 2x2 matrices B such that B[1 -1]^t=0 vector The 0 matrix is in this set because the matrix 2x2 consisting of all 0s multiplied by [1 -1]^t is =0. Now I want to make sure I'm correctly applying the latter 2 conditions. If U and V are in this set, then the following is true. If U*[1 -1]^t=0 and V*[1 -1]^t=0 U+V=(U+V)[1 -1]^t. Since U and V are 0, U+V=0. thus U+V=(0+0)[1 -1]^t.=0, thus U+V is in B? For condition 3: if V1*[1 -1]^t=0 is in the set, then cV must be in the set for it to be a subspace. cV1*[1 -1]^t=c*0*[1 -1]^t=0, thus cV is in the set? Thus the set of 2x2 matrices B such that B[1 -1]^t=0 vector is a subspace.