document.write( "Question 891936: Solve the following system equation using inverse matrix.
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Algebra.Com's Answer #540247 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( " x +  y +  z = 2\r\n" );
document.write( "3x + 2y – 2z = 8\r\n" );
document.write( "2x – 3y – 4z = 0\r\n" );
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document.write( "By matrix inversion.\r\n" );
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document.write( "I will assume you already know how to find the inverse\r\n" );
document.write( "of a matrix, and how to multiply two matrices. If you don't, \r\n" );
document.write( "post again asking how.\r\n" );
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document.write( "First we form three matrices, A, X, and B.\r\n" );
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document.write( "1. Matrix A is the 3x3 coefficient matrix A, which consists \r\n" );
document.write( "of just the three columns of x, y, and z coefficients. in \r\n" );
document.write( "that order, but does not contain the column of constants.\r\n" );
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document.write( "\"A=%28matrix%283%2C3%2C1%2C1%2C1%2C3%2C2%2C-2%2C2%2C-3%2C-4%29%29\". \r\n" );
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document.write( "2. Matrix X is the 3x1 matrix of variables \"X=%28matrix%283%2C1%2Cx%2Cy%2Cz%29%29\"\r\n" );
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document.write( "3. Matrix B is the 3x1 matrix, whose only column is the\r\n" );
document.write( "column of constants: \"%28matrix%283%2C1%2C2%2C8%2C0%29%29\"\r\n" );
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document.write( "Next we form the matrix equation:\r\n" );
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document.write( "       \"A%2AX+=+B\"\r\n" );
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document.write( "or\r\n" );
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document.write( "To solve the equation\r\n" );
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document.write( "       \"A%2AX+=+B\"\r\n" );
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document.write( "we left-multiply both sides by \"A%5E%28-1%29\", the inverse of \"A\".\r\n" );
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document.write( " \"A%5E%28-1%29%2A%28A%2AX%29+=+A%5E%28-1%29%2AB\"\r\n" );
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document.write( "Then since the associatitive principle holds for matrix multiplication,\r\n" );
document.write( "(even though the commutative principle DOES NOT!!!), we can move\r\n" );
document.write( "the parentheses on the left around the first two matrix factors:\r\n" );
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document.write( "\"%28A%5E%28-1%29%2AA%29%2AX+=+A%5E%28-1%29%2AB\"\r\n" );
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document.write( "Now since \"%28A%5E%28-1%29%2AA%29=I\", where I is the identity matrix, the\r\n" );
document.write( "above becomes:\r\n" );
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document.write( "\"I%2AX+=+A%5E%28-1%29%2AB\"\r\n" );
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document.write( "and by the identity property:\r\n" );
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document.write( "\"X=A%5E%28-1%29%2AB\"\r\n" );
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document.write( "Performing these operations with the actual matrices we have\r\n" );
document.write( "the equation \"AX=B\"\r\n" );
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document.write( "Next we form the inverse of A, which is written A-1.\r\n" );
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document.write( "Then we indicate the left multiplication of both sides by\r\n" );
document.write( "\"A%5E%28-1%29\" to get the equation \"A%5E%28-1%29%28A%2AX%29=A%5E%28-1%29B\":\r\n" );
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document.write( "Next we use the associative principle to move the parentheses so that\r\n" );
document.write( "they are around the first two factors to get the equation \"%28A%5E%28-1%29%2AA%29%2AX=A%5E%28-1%29%2AB\":\r\n" );
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document.write( "Now we perform the actual multiplications and we get the equation \"IX=A%5E%28-1%29%2AB\":\r\n" );
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document.write( "Then when we multiply the identity matrix \"I\" by the column matrix of\r\n" );
document.write( "variables, we just get the matrix of variables, or the \r\n" );
document.write( "equation \"X=A%5E%28-1%29B\"\r\n" );
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document.write( "\"%28matrix%283%2C1%2Cx%2Cy%2Cz%29%29=%28matrix%283%2C1%2C20%2F19%2C32%2F19%2C-14%2F19%29%29\"\r\n" );
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document.write( "------------------\r\n" );
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document.write( "Checking:\r\n" );
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document.write( "x + y + z = 2\r\n" );
document.write( "\"20%2F19%2B32%2F19-14%2F19=2\"\r\n" );
document.write( "\"38%2F19=2\"\r\n" );
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document.write( "That checks.\r\n" );
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document.write( "3x + 2y – 2z = 8\r\n" );
document.write( "\"3%2820%2F19%29%2B2%2832%2F19%29-2%28-14%2F19%29=8\"\r\n" );
document.write( "\"60%2F19%2B64%2F19%2B28%2F19=8\"\r\n" );
document.write( "\"152%2F19=8\"\r\n" );
document.write( "\"8=8\"\r\n" );
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document.write( "That checks.\r\n" );
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document.write( "2x – 3y – 4z = 0\r\n" );
document.write( "\"2%2820%2F19%29-3%2832%2F19%29-4%28-14%2F19%29=0\"\r\n" );
document.write( "\"40%2F19-96%2F19%2B56%2F19=0\"\r\n" );
document.write( "\"0%2F19=0\"\r\n" );
document.write( "\"0=0\"\r\n" );
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document.write( "That checks.\r\n" );
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document.write( "Terrible answers, but they're correct.  Did you copy the problem\r\n" );
document.write( "correctly?  It's the correct solution for the system you posted.\r\n" );
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document.write( "Edwin
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