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標題:
Matrix
發問:
A and B are two different 2 x 2 matrices. If A is not the inverse of B, B is not the inverse of A, A is not equal to the Identity matrix I and B is also not equal to I, can AB = BA? If yes, please give example, thanks.
最佳解答:
If A = (a11, a12 ... a21, a22), B = (b11, b12 ... b21, b22) then AB = (a11b11+a12b21, a11b12+a12b22 ... a21b11+a22b21, a21b12+a22b22) and BA = (b11a11+b12a21, b11a12+b12a22 ... b21a11+b22a21, b21a12+b22a22) So, if AB = BA, then a11b11+a12b21 = b11a11+b12a21 a11b12+a12b22 = b11a12+b12a22 a21b11+a22b21 = b21a11+b22a21 a21b12+a22b22 = b21a12+b22a22 That is, a12b21 = b12a21 a11b12+a12b22 = b11a12+b12a22 a21b11+a22b21 = b21a11+b22a21 a21b12 = b21a12 In order to make these four equations correct, a simplest way is to set most of the variables to become zero (but not all, as it is means A, B are zero matrix). So, for example, only set a12, b12 not zero, and others are all zero will be OK. That is, A = (0, 2 ... 0, 0), B = (0, 3 ... 0, 0) then AB = BA = zero matrix You may say that A, B are two similar matrices in this example. Ok, how about set a12, b11 not zero? That is A = (0,2 ... 0, 0), B = (3, 0 ... 0, 0) For more complicated, that is A, B are not zero matrix. A = (1, 3 ... 6, 1), B = (1, 4 ... 8, 1) so, AB = BA = (25, 7 ... 14, 25)
Matrix
發問:
A and B are two different 2 x 2 matrices. If A is not the inverse of B, B is not the inverse of A, A is not equal to the Identity matrix I and B is also not equal to I, can AB = BA? If yes, please give example, thanks.
最佳解答:
If A = (a11, a12 ... a21, a22), B = (b11, b12 ... b21, b22) then AB = (a11b11+a12b21, a11b12+a12b22 ... a21b11+a22b21, a21b12+a22b22) and BA = (b11a11+b12a21, b11a12+b12a22 ... b21a11+b22a21, b21a12+b22a22) So, if AB = BA, then a11b11+a12b21 = b11a11+b12a21 a11b12+a12b22 = b11a12+b12a22 a21b11+a22b21 = b21a11+b22a21 a21b12+a22b22 = b21a12+b22a22 That is, a12b21 = b12a21 a11b12+a12b22 = b11a12+b12a22 a21b11+a22b21 = b21a11+b22a21 a21b12 = b21a12 In order to make these four equations correct, a simplest way is to set most of the variables to become zero (but not all, as it is means A, B are zero matrix). So, for example, only set a12, b12 not zero, and others are all zero will be OK. That is, A = (0, 2 ... 0, 0), B = (0, 3 ... 0, 0) then AB = BA = zero matrix You may say that A, B are two similar matrices in this example. Ok, how about set a12, b11 not zero? That is A = (0,2 ... 0, 0), B = (3, 0 ... 0, 0) For more complicated, that is A, B are not zero matrix. A = (1, 3 ... 6, 1), B = (1, 4 ... 8, 1) so, AB = BA = (25, 7 ... 14, 25)
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