document.write( "Question 243147: hello \r
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\n" ); document.write( "find the values of x, y, and z by i)cramers rule ii)matrix inversion
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Algebra.Com's Answer #178188 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
hello
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\n" ); document.write( "\n" ); document.write( "find the values of x, y, and z by i)cramers rule ii)matrix inversion
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document.write( "Cramer's rule:\r\n" );
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document.write( "I will assume you already know how to get the value of a 3x3\r\n" );
document.write( "determinant.  If you don't know how, post again asking how\r\n" );
document.write( "it's done.\r\n" );
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document.write( "There are 4 columns,\r\n" );
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document.write( "1. The column of x-coefficients \"matrix%283%2C1%2C3%2C-1%2C4%29\"\r\n" );
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document.write( "2. The column of y-coefficients \"matrix%283%2C1%2C-2%2C1%2C3%29\"\r\n" );
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document.write( "3. The column of z-coefficients \"matrix%283%2C1%2C1%2C-2%2C2%29\" \r\n" );
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document.write( "4. The column of constants:     \"matrix%283%2C1%2C-12%2C10%2C-1%29\"\r\n" );
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document.write( "There are four determinants:\r\n" );
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document.write( "1. The determinant \"D\" consists of just the three columns\r\n" );
document.write( "of x, y, and z coefficients. in that order, but does not\r\n" );
document.write( "contain the column of constants.\r\n" );
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document.write( "\"D=abs%28matrix%283%2C3%2C3%2C-2%2C1%2C-1%2C1%2C-2%2C4%2C3%2C2%29%29\". \r\n" );
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document.write( "It has value \"D=29\". \r\n" );
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document.write( "2. The determinant \"D%5Bx%5D\" is like the determinant \"D\"\r\n" );
document.write( "except that the column of x-coefficients is replaced by the\r\n" );
document.write( "column of constants.  \"D%5Bx%5D\" does not contain the column \r\n" );
document.write( "of x-coefficients.\r\n" );
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document.write( "\"D%5Bx%5D=abs%28matrix%283%2C3%2C-12%2C-2%2C1%2C10%2C1%2C-2%2C-1%2C3%2C2%29%29\".\r\n" );
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document.write( "It has value \"D%5Bx%5D=-29\".\r\n" );
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document.write( "3. The determinant \"D%5By%5D\" is like the determinant \"D\"\r\n" );
document.write( "except that the column of y-coefficients is replaced by the\r\n" );
document.write( "column of constants.  \"D%5By%5D\" does not contain the column \r\n" );
document.write( "of y-coefficients.\r\n" );
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document.write( "\"D%5By%5D=abs%28matrix%283%2C3%2C3%2C-12%2C1%2C-1%2C10%2C-2%2C4%2C-1%2C2%29%29\".\r\n" );
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document.write( "It has value \"D%5By%5D=87\".\r\n" );
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document.write( "4. The determinant \"D%5Bz%5D\" is like the determinant \"D\"\r\n" );
document.write( "except that the column of z-coefficients is replaced by the\r\n" );
document.write( "column of constants.  \"D%5Bz%5D\" does not contain the column \r\n" );
document.write( "of z-coefficients.\r\n" );
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document.write( "\"D%5Bz%5D=abs%28matrix%283%2C3%2C3%2C-2%2C12%2C-1%2C1%2C10%2C4%2C3%2C-1%29%29\".\r\n" );
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document.write( "It has value \"D%5Bx%5D=-87\".\r\n" );
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document.write( "Now the formulas for x, y and z are\r\n" );
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document.write( "\"x=D%5Bx%5D%2FD=-29%2F29=-1\"\r\n" );
document.write( "\"y=D%5By%5D%2FD=87%2F29=3\"\r\n" );
document.write( "\"x=D%5Bz%5D%2FD=-87%2F3=-3\"\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%2C3%2C-2%2C1%2C-1%2C1%2C-2%2C4%2C3%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: \"%28matrix%283%2C1%2C-12%2C10%2C-1%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%2C-1%2C3%2C-3%29%29\"\r\n" );
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document.write( "Edwin
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