More Martrices Questions?

Question

True or False Questions:
1. All Square Marcrices Have Multiplicative Identities, T or F
2. Only square matrices have multiplicative inverses, T or F
3. Some square matrices do not have mutliplicative inverses, T or F
4. Some square matrices do not have multiplicative identities, T or F
5. All identity matrices are square matrices, T or F

I know some are repetitive, but if you could answer them all that would be so great =D

Answer 1

An identity matrix must be square, so 5 is true. Inverses must multiply to the same answer in either order, so they must be square, so 2 is true.

Any square matrix multiplied by its identity will equal itself, so 4 is false and 1 is true.

Inverses must multiply to come out the identity which is impossible if one matrix is all zeros, so 3 is true.

Answer 2

1. T.
2. T. Some books define the inverse of a matrix as the transpose of the adjoint divided by the determinant. Division by zero occurs when the determinant is zero (a singular matrix). So you can take the determinant of square matrices only.
3. T.
4. F. I_n is the multiplicative identity of an n by n matrix
5. T.

Answer 3

1) is true. The MxM identity matrix is an Multiplicative Identitiy of an MxM square matrix.
2) is true. Only if A and B are square matrices, can you mulitply both AB and BA, a requirement if B is the inverse of A.
3) is true.
|0 0|
|0 0| for example has no mutiplicative inverse. Only those MxM matrices with rank(M) have an inverse.
4) Is not true. The MxM identity matrix, a diagonal matrix of 1s, is an Multiplicative Identitiy of an MxM square matrix. IA = AI = A, for any square matrix A.
5) True. By definition the identity matrix is square.