Which of the following subsets of the vector space Mnn are subspaces?
(a) The set of all n × n symmetric matrices
(b) The set of all n × n diagonal matrices
(c) The set of all n × n nonsingular matrices
plz how this solve
To determine whether a subset of a vector space is a subspace, we need to check three conditions: closure under addition, closure under scalar multiplication, and containing the zero vector.
(a) The set of all n × n symmetric matrices:
To check closure under addition, we need to ensure that if A and B are symmetric matrices, then their sum A + B is also a symmetric matrix. To verify this, we consider the transpose of A + B:
(A + B)^T = A^T + B^T.
Since A and B are symmetric matrices, we have A^T = A and B^T = B. Therefore, (A + B)^T = A + B, which implies that A + B is also symmetric. Thus, this subset is closed under addition.
Next, we check closure under scalar multiplication. Given a symmetric matrix A, we need to show that if c is a scalar, then cA is also symmetric. We have:
(cA)^T = cA^T.
Since A is symmetric, we have A^T = A. Therefore, (cA)^T = cA. This implies that cA is symmetric. Thus, this subset is closed under scalar multiplication.
Lastly, we need to ensure that the zero vector is in this subset. The zero matrix is a symmetric matrix, so it satisfies this condition.
Therefore, the set of all n × n symmetric matrices is a subspace of the vector space Mnn.
(b) The set of all n × n diagonal matrices:
Similar to the previous case, we check closure under addition by considering two diagonal matrices A and B:
(A + B)^T = A^T + B^T.
Since the transpose of a diagonal matrix is itself, we have (A + B)^T = A + B. Therefore, A + B is also a diagonal matrix, and this subset is closed under addition.
Next, let A be a diagonal matrix, and c be a scalar. We need to show that cA is also a diagonal matrix:
(cA)^T = cA^T.
Since A is a diagonal matrix, its transpose is itself, i.e., A^T = A. Then, (cA)^T = cA. Thus, cA is also a diagonal matrix, and this subset is closed under scalar multiplication.
The zero vector, which is the matrix with all entries equal to zero, is a diagonal matrix. Therefore, this subset contains the zero vector.
Hence, the set of all n × n diagonal matrices is a subspace of the vector space Mnn.
(c) The set of all n × n nonsingular matrices:
A nonsingular matrix is a square matrix whose determinant is non-zero. To determine if this subset is a subspace, we need to evaluate the three conditions: closure under addition, closure under scalar multiplication, and containing the zero vector.
First, consider two nonsingular matrices A and B. It is known that the sum of two nonsingular matrices may not necessarily be nonsingular. Therefore, this subset does not satisfy closure under addition.
Similarly, scalar multiplication does not preserve the property of being nonsingular. Multiplying a nonsingular matrix by a scalar can result in a matrix with a zero determinant. Hence, this subset does not satisfy closure under scalar multiplication.
Furthermore, the zero vector, which is the matrix with all entries equal to zero, is singular since its determinant is zero. Therefore, this subset does not contain the zero vector.
In conclusion, the set of all n × n nonsingular matrices is not a subspace of the vector space Mnn.