document.write( "Question 620071: Solve the system using the inverse method.\r
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Algebra.Com's Answer #390542 by Edwin McCravy(20063)\"\" \"About 
You can put this solution on YOUR website!
x + 2y + 3z = -2
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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%2C2%2C3%2C1%2C1%2C1%2C-1%2C1%2C2%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:  B = \"%28matrix%283%2C1%2C-2%2C-7%2C-10%29%29\"\r\n" );
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document.write( "Next we form the matrix equation:\r\n" );
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document.write( "                               AX = 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( "                               AX = B\r\n" );
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document.write( "we left-multiply both sides by A-1, the inverse of A.\r\n" );
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document.write( "                          A-1(AX) = A-1B\r\n" );
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document.write( "Then since the associative 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( "                          (A-1A)X = A-1B\r\n" );
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document.write( "Now since A-1A = I, where I is the identity matrix, the\r\n" );
document.write( "above becomes:\r\n" );
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document.write( "                              IX = A-1B\r\n" );
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document.write( "and by the identity property:\r\n" );
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document.write( "                               X = A-1B\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 find the inverse of A, which is written A-1.\r\n" );
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document.write( "A-1 = \"%28matrix%283%2C3%2C1%2C+-1%2C+-1%2C+-3%2C+5%2C+2%2C+2%2C+-3%2C+-1%29%29%0D%0A\"\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-1(A*X) = A-1B:\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 \r\n" );
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document.write( "                (A-1A)X=A-1B:\r\n" );
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document.write( "\"A%5E%28-1%29\" to get the equation (A-1A)X = A-1B:\r\n" );
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document.write( "When we perform the matrix multiplication we get:\r\n" );
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document.write( "The matrix on the left is the identity matrix\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-1B\r\n" );
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document.write( "\"%28matrix%283%2C1%2Cx%2Cy%2Cz%29%29=%28matrix%283%2C1%2C15%2C-49%2C27%29%29\"\r\n" );
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document.write( "or x=15, y=-49, z=27\r\n" );
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document.write( "Edwin
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