Matrix multiplication
Multiplying two matrices: the condition on dimensions, the row-by-column calculation, non-commutativity, the identity matrix, powers and the product of a matrix by a column vector.
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A few cards from the deck
Matrix product: what is it?
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For A of dimensions m × p and B of dimensions p × n, the m × n matrix whose entry (i, j) is the sum, for k from 1 to p, of a_ik × b_kj.
You multiply row i of A by column j of B, entry by entry, then add. This requires A to have as many columns as B has rows.
With A = (1 2; 3 4) and B = (0 1; 5 −1), what is BA?
(3 4; 2 6)(10 −1; 20 −1)(0 2; 15 −4)(3 2; 4 6)
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(3 4; 2 6)For BA, take the rows of B and the columns of A: 0×1 + 1×3 = 3, 0×2 + 1×4 = 4, 5×1 − 1×3 = 2, 5×2 − 1×4 = 6. (10 −1; 20 −1) is AB, (0 2; 15 −4) the entry-by-entry product, and (3 2; 4 6) the transpose of the right answer.
The product AB exists if the number of … of A equals the number of … of B.
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The product AB exists if the number of columns of A equals the number of rows of B.
Each sum pairs a row of A with a column of B, entry by entry: the two must have the same number of entries.
Non-commutativity: what is it?
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The property of matrix multiplication that, in general, AB and BA are different, and one may exist without the other.
With A = (1 2; 3 4) and B = (0 1; 5 −1), AB = (10 −1; 20 −1) whereas BA = (3 4; 2 6). So you can never swap the factors without justifying it.
Does AB = 0 (the zero matrix) imply A = 0 or B = 0?
- Yes, as for real numbers: a zero product always has a zero factor
- No, even for square matrices: two nonzero factors can give a zero product
- Yes, because det A × det B = 0 forces one of the two matrices to be zero
- No, but only when A and B are not square: for square ones, yes
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No, even for square matrices: two nonzero factors can give a zero product
(1 0; 0 0)(0 0; 0 1) is the zero matrix although neither square factor is. A zero determinant means "not invertible", not "zero matrix". As a result you cannot cancel A in AB = AC to get B = C.