Proving that if AB = 0, then either A or B is a zero matrix

If AB = 0, then either A or B is a zero matrix.

To prove that if AB = 0, then either A or B is a zero matrix, we will consider two cases

To prove that if AB = 0, then either A or B is a zero matrix, we will consider two cases.

Case 1: A is not a zero matrix.
If A is not a zero matrix, then there exists at least one entry in A that is non-zero. Let’s assume that the entry at position (i, j) in A is non-zero.

Now, since AB = 0, we can write the product AB as follows:
(AB)ij = Σ(Aik * Bkj) = 0 for all i, j

Consider the term Aik * Bkj in the summation. For this term to contribute to the value of (AB)ij, both Aik and Bkj must be non-zero. However, we assumed that Aik is non-zero, so for (AB)ij to be zero, Bkj must be zero for all k.

This implies that every entry in the jth column of B must be zero, as changing the row index i does not affect Bkj. Therefore, B is a zero matrix.

Case 2: B is not a zero matrix.
If B is not a zero matrix, then there exists at least one entry in B that is non-zero. Let’s assume that the entry at position (k, l) in B is non-zero.

Similarly to Case 1, we can write the product AB as follows:
(AB)ij = Σ(Aik * Bkj) = 0 for all i, j

Consider the term Aik * Bkj in the summation. For this term to contribute to the value of (AB)ij, both Aik and Bkj must be non-zero. However, we assumed that Bkj is non-zero, so for (AB)ij to be zero, Aik must be zero for all i.

This implies that every entry in the kth row of A must be zero, as changing the column index j does not affect Aik. Therefore, A is a zero matrix.

Since we have considered both cases, we can conclude that if AB = 0, then either A or B is a zero matrix.

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