Determine which of the following are subspaces of Mnn. (a)...

Determine which of the following are subspaces of Mnn. (a) The set of all upper triangular matrices. (b) The set of all n x n matrices A such that det(A)=0. (c) The set of all n x n matrices A such that tr(A)=0. (d) The set of all n x n matrices A such that AT=-A. (e) The set of all n x n matrices A for which Ax=0 has only the trivial solution. (f) The set of all n x n matrices A such that AB=BA for some fixed n x n matrix B. A. a, b, c, e B. a, c, d, e C. a, b, c, d, f D. a, c, d, f

Transcript

00:01 So in this problem we are given some subsets of n cross n matrices and we need to see which of them is a subspace so we will verify the following two properties so if v is a vector space and w is any subset of v non -entry then w is subspace of v so w is a subspace of the if it is closed under addition closed under addition that is w1 w2 in w implies that w1 plus w2 is in w and the second one closed under scalar multiplication closed under scalar multiplication that is alpha is in r w is in w implies alpha w is in w so we will verify these two properties so the first is upper triangular matrices upper triangular matrices okay so see that any upper triangular matrix will be like which has this and this is zero if i take the addition something like this i'll be getting since the addition is like component wise so this the below will be zero and this will be the sum of so if this is star 1 this is star 2 so i have to take the sum of star 1 plus star 2 so but it is still an upper triangular matrix and similarly if i multiply anything with alpha this thing then what happened since alpha is getting multiply each component so below diagonal element will still be 0 and this will be multiplied by alpha so therefore this is a vector space vector subspace okay coming to the next one so the next one it says that the n cross and matrices such that the determinant of is c so this is not a vector subspace so one like one easy example is if i take a to be the identity matrix which has determinant equals to 1 so hence it's non -zero but if i'll multiply with 0 so this is a scalar multiplication so what is this is 0 and what is the determinant it is 0 so this is not in the space so this implies this is not a subspace okay so this is not what is the third the third says that the trace of a is zero so so we will we know that a is a matrix then and b is a matrix then we know that trace of a plus b is trace of a plus trace of b now if a and b are in w which is like what is w w is the set of all matrices which trace is the trace of a is 0 and trace of b is 0 and hence trace of a plus b is 0 and…

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