- #1
blueman11
- 1
- 0
Show that if A and B are square matrices of the same size such that B is an
invertible matrix, then A must be a zero matrix.
invertible matrix, then A must be a zero matrix.
We need information about how A and B are related. ie. AB = 0 or something.blueman11 said:Show that if A and B are square matrices of the same size such that B is an
invertible matrix, then A must be a zero matrix.
To prove that A is a zero matrix when B is invertible and the same size as A, we can use the property that if the product of two matrices is a zero matrix, then at least one of the matrices must be a zero matrix. Since B is invertible, it cannot be a zero matrix. Therefore, A must be the zero matrix.
A zero matrix is a matrix where all elements are equal to zero. It is also known as a null matrix or a matrix of all zeros.
If B is not invertible, then it is possible for the product of B and A to be a zero matrix even if A is not a zero matrix. This would invalidate the proof that A is a zero matrix based on the property of matrix multiplication.
Yes, A can be a non-square matrix as long as it is the same size as B and follows the property that if the product of A and B is a zero matrix, then at least one of the matrices must be a zero matrix.
Yes, another way to prove this is by using the fact that if B is invertible, then it has a unique inverse. If we take the inverse of B and multiply it by the product of A and B, we should get the zero matrix. This would mean that the only possible solution for A is the zero matrix.