document.write( "Question 347597: 1.(a)given the matrix=A
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Algebra.Com's Answer #248469 by Edwin McCravy(20059) $\"\"$ $\"About$

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1. (a) Given the matrix $\"A=%28matrix%283%2C2%2C2%2C3%2C0%2C-1%2C1%2C4%29%29\"$ ,
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document.write( "Dimensions are \"down by across\".  It is 3 elements down\r\n" );
document.write( "and 2 elements across.  So it's a 3 by 2.\r\n" );
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document.write( "No they can't, because the matrix on the left is a 3 by 1 matrix and the\r\n" );
document.write( "matrix on the right is a 3 by 2 matrix.\r\n" );
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document.write( "You can only multiply a \"p by q\" matrix on the left by a \"q by r\" matrix \r\n" );
document.write( "on the right and get a \"p by r\" matrix.\r\n" );
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document.write( "        (p by q)×(q by r) = (p by r)\r\n" );
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document.write( "The \"inner\" dimensions q must be the same, and if so, then the product  \r\n" );
document.write( "matrix will have the \"outer\" dimensions, p by r.  Examples:\r\n" );
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document.write( "A 6 by 5 matrix on the left can be multiplied by a 5 by 9 matrix on the right\r\n" );
document.write( "to give a 6 by 9 matrix, because the inner dimensions are both 5 and the outer\r\n" );
document.write( "dimensions are 6 by 9.\r\n" );
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document.write( "        (6 by 5)×(5 by 9) = (6 by 9)\r\n" );
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document.write( "A 3 by 1 matrix on the left can be multiplied by a 1 by 3 matrix on the right\r\n" );
document.write( "to give a 3 by 3 matrix, because the inner dimensions are both 1 and the outer\r\n" );
document.write( "dimensions are 3 by 3. \r\n" );
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document.write( "        (3 by 1)×(1 by 3) = (3 by 3)\r\n" );
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document.write( "A 1 by 3 matrix on the left can be multiplied by a 3 by 1 matrix on the right\r\n" );
document.write( "to give a 1 by 1 matrix, because the inner dimensions are both 3 and the outer\r\n" );
document.write( "dimensions are 1 by 1.\r\n" );
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document.write( "        (1 by 3)×(3 by 1) = (1 by 1)\r\n" );
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document.write( "A 58 by 29 matrix on the left can be multiplied by a 29 by 73 matrix on the\r\n" );
document.write( "right to give a 58 by 73 matrix, because the inner dimensions are both 29 and\r\n" );
document.write( "the outer dimensions are 58 by 73.\r\n" );
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document.write( "        (58 by 29)×(29 by 73) = (58 by 73)\r\n" );
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document.write( "However, in your problem (b), a 3 by 1 matrix on the left cannot be multiplied\r\n" );
document.write( "by a 3 by 2 matrix on the right, because the inner dimensions are 1 and 3 are\r\n" );
document.write( "they are not the same. \r\n" );
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document.write( "        (3 by 1)×(3 by 2)\r\n" );
document.write( "is not defined because the inner dimensions are not the same.\r\n" );
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document.write( "If you had been given these matrices instead:\r\n" );
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document.write( "then you would have had a 1 by 3 matrix on the left to multiply by a 3 by 2\r\n" );
document.write( "matrix on the right, and they could be multiplied because the inner dimensions\r\n" );
document.write( "are both 3.  The product matrix would be a 1 by 2 matrix because they would be\r\n" );
document.write( "the outer dimensions.\r\n" );
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document.write( "        (1 by 3)×(3 by 2) = (1 by 2)\r\n" );
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document.write( "However if you wrote the second one on the left and the first one on the right\r\n" );
document.write( "like this:\r\n" );
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document.write( "you could not multiply them because the inner dimensions would not be the same:\r\n" );
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document.write( "        (3 by 2)×(1 by 3 is not defined.  \r\n" );
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document.write( "The commutative principle does not hold for matrix multiplication like it\r\n" );
document.write( "holds for ordinary number multiplication.\r\n" );
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document.write( "That is when you multiply 2×3 you get 6, which is the same as when you\r\n" );
document.write( "multiply 3×2.  So the commutative principle holds for ordinary number\r\n" );
document.write( "multiplication.  \r\n" );
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document.write( "Notice this however.  When you multiply \r\n" );
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document.write( "and they can be multiplied, because\r\n" );
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document.write( "        (2 by 2)×(2 by 2) = (2 by 2)\r\n" );
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document.write( "the inner dimensions are the same,\r\n" );
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document.write( "you get this 2 by 2 matrix  \r\n" );
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document.write( "But when you multiply \r\n" );
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document.write( "and they can be multiplied, too, because\r\n" );
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document.write( "        (2 by 2)×(2 by 2) = (2 by 2)\r\n" );
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document.write( "the inner dimensions are the same,\r\n" );
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document.write( "you get this 2 by 2 matrix  \r\n" );
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document.write( "So if you reverse the order of two matrices, even if they can still be \r\n" );
document.write( "multiplied, you don't get the same answer.\r\n" );
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document.write( "Edwin

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