Let P = [[1, 1, -1], [-1, -2, -3], [-3, 2, -4], [1, 5, 1]] be a 4x3 matrix and Q = [[-9, -6, 3, -1], [5, 10, 20, 5]] be ...

Question

Let P = [[1, 1, -1], [-1, -2, -3], [-3, 2, -4], [1, 5, 1]] be a 4x3 matrix and Q = [[-9, -6, 3, -1], [5, 10, 20, 5]] be a 2x4 matrix (wait - see below for actual matrices). Let P = [[1, 1, -1], [-1, -1, -3], [-3, 2, 3]] and Q = [[5, 10, 12, 6], [10, 20, 5, 5]] -- based on image.

Actual question: Let P = [[1, 1, -1], [-1, -3, 4], [-1, 0, 2], [1, -2, 3]] and Q = [[-9, -6, -3, -1], [5, 10, 20, 5]] be two matrices. Then the rank of P * Q is ___________.

Accepted Answer

2 The rank of a product of matrices satisfies rank(PQ) <= min(rank(P), rank(Q)). Here P is 4x3, so rank(P) <= 3, and Q is 2x4, so rank(Q) <= 2. The product PQ is 4x2 (wait, dimensions don't match for a 4x3 times 2x4). Actually for P (mxn) times Q (nxp), we need the inner dimensions to match. Let me reconsider: P is likely a 3x4 matrix and Q is a 4x2 matrix, or P is 4x2 and Q is 2x3. The rank of PQ <= min(rank(P), rank(Q)) <= min of the inner dimensions. Since the answer is 2, both P and Q achieve rank 2.