document.write( "Question 186733: Ok, so I am given 4 matrices and 2 operations to perform (if possible). I don't think the operations can be performed, but would like to verify that that's true.\r
\n" ); document.write( "\n" ); document.write( "Here are the 4 matrices:
\n" ); document.write( "A =
\n" ); document.write( "[2 -1 3]
\n" ); document.write( "[0 4 -2]\r
\n" ); document.write( "\n" ); document.write( "B =
\n" ); document.write( "[-3 1]
\n" ); document.write( "[ 2 5]
\n" ); document.write( "
\n" ); document.write( "C =
\n" ); document.write( "[-1 0 2]
\n" ); document.write( "[ 4 -3 1]
\n" ); document.write( "[-2 3 5]\r
\n" ); document.write( "\n" ); document.write( "D =
\n" ); document.write( "[3 -2]
\n" ); document.write( "[0 -1]
\n" ); document.write( "[1 2]\r
\n" ); document.write( "\n" ); document.write( "Here are the 2 operations:
\n" ); document.write( "1. (B^5)A + D (I think the addition of matrices requires identical
\n" ); document.write( " dimensions & here we have a 2x2 added to a 3x2)\r
\n" ); document.write( "\n" ); document.write( "2. (3)BA + (4)AC (I think the multiplication of matrices requires the
\n" ); document.write( " number of rows in the first matrix to match the number
\n" ); document.write( " of columns in the second. Here we have B as a 2x2 and
\n" ); document.write( " A as a 2x3; then, A as a 2x3 and C as a 3x3)\r
\n" ); document.write( "\n" ); document.write( "Thanks for taking a look, I appreciate your assistance. \n" ); document.write( "
Algebra.Com's Answer #140030 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
Ok, so I am given 4 matrices and 2 operations to perform (if possible). I don't think the operations can be performed, but would like to verify that that's true.\r
\n" ); document.write( "\n" ); document.write( "Here are the 4 matrices:
\n" ); document.write( "A =
\n" ); document.write( "[2 -1 3]
\n" ); document.write( "[0 4 -2]\r
\n" ); document.write( "\n" ); document.write( "B =
\n" ); document.write( "[-3 1]
\n" ); document.write( "[ 2 5]
\n" ); document.write( "
\n" ); document.write( "C =
\n" ); document.write( "[-1 0 2]
\n" ); document.write( "[ 4 -3 1]
\n" ); document.write( "[-2 3 5]\r
\n" ); document.write( "\n" ); document.write( "D =
\n" ); document.write( "[3 -2]
\n" ); document.write( "[0 -1]
\n" ); document.write( "[1 2]\r
\n" ); document.write( "\n" ); document.write( "Here are the 2 operations:
\n" ); document.write( "1. (B^5)A + D (I think the addition of matrices requires identical
\n" ); document.write( " dimensions & here we have a 2x2 added to a 3x2)\r
\n" ); document.write( "\n" ); document.write( "2. (3)BA + (4)AC (I think the multiplication of matrices requires the
\n" ); document.write( " number of rows in the first matrix to match the number
\n" ); document.write( " of columns in the second. Here we have B as a 2x2 and
\n" ); document.write( " A as a 2x3; then, A as a 2x3 and C as a 3x3)\r
\n" ); document.write( "\n" ); document.write( "Thanks for taking a look, I appreciate your assistance.
\n" ); document.write( "
\r\n" );
document.write( "The rules are \r\n" );
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document.write( "1. For adding.  We can only add a pxq matrix to another pxq matrix\r\n" );
document.write( "and when we do, we end up with another pxq matrix.\r\n" );
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document.write( "2. For multiplying.  We can only multiply a pxq matrix times a qxr\r\n" );
document.write( "matrix and end up with a pxr matrix.\r\n" );
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document.write( "A is a 2x3 matrix\r\n" );
document.write( "B is a 2x2 matrix\r\n" );
document.write( "C is a 3x3 matrix\r\n" );
document.write( "D is a 3x2 matrix\r\n" );
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document.write( "For \"%28B%5E5%29A+%2B+D\"\r\n" );
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document.write( "The first factor in the first term is:\r\n" );
document.write( "\"B%5E5\" \r\n" );
document.write( "That's a 2x2 times a 2x2 times a 2x2 times a 2x2 times a 2x2.\r\n" );
document.write( "So we end up with another 2x2.\r\n" );
document.write( "\r\n" );
document.write( "Now the whole first term is:\r\n" );
document.write( "\"%28B%5E5%29A\"\r\n" );
document.write( "So here we are multiplying a 2x2 times A, which is a 2x3.\r\n" );
document.write( "This can be multiplied and we will get a 2x3.\r\n" );
document.write( "Now we must add this to D, which is a 3x2. Uh oh!!\r\n" );
document.write( "We can't add these, so that's undefined.  You are right.\r\n" );
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document.write( "------\r\n" );
document.write( "But I think you're wrong about the second one.\r\n" );
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document.write( "\"%283%29BA+%2B+%284%29AC\"\r\n" );
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document.write( "The constants (3) and (4) in front do not affect the dimensions.\r\n" );
document.write( "So we can ignore them to find out if this is defined and, if so,\r\n" );
document.write( "what kind of matrix we will end up with.\r\n" );
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document.write( "The first term involves chiefly BA.  That's a 2x2 times a 2x3.\r\n" );
document.write( "When that is multiplied, we will get a 2x3.\r\n" );
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document.write( "The second term involves chiefly AC.  That's a 2x3 times a 3x3.\r\n" );
document.write( "When that is multiplied, we will get a 2x3.\r\n" );
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document.write( "So we can add these because they have the same dimensions, and\r\n" );
document.write( "we will end up with another 2x3 matrix.\r\n" );
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document.write( "The answer to the second problem is this 2x3 matrix\r\n" );
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document.write( "[-66  69  39]\r\n" );
document.write( "[ 92 -18 -36]\r\n" );
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document.write( "If you have trouble getting this answer, you may email me.\r\n" );
document.write( "AnlytcPhil@aol.com and I'll help you get it.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
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