document.write( "Question 1153962: Consider the following system of linear equations.
\n" ); document.write( "3x − 2y + z = 4
\n" ); document.write( "x + y − z = 2
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\n" ); document.write( "(i) Write the system in the form Ax = b.
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\n" ); document.write( "(iv) Find the inverse of A using the determinant and the adjoint of A, i.e.,
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\n" ); document.write( "(v) Comment on your answers in part (iii) and (iv).
\n" ); document.write( "(vi) Solve the system using the inverse matrix found in part (iv). \n" ); document.write( "
Algebra.Com's Answer #776344 by Edwin McCravy(20064)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We write the system it in the form AX = B,\r\n" );
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document.write( "Next we show that the determinant of the matrix A is not 0, which\r\n" );
document.write( "tells us that the matrix A has an inverse A-1. We expand\r\n" );
document.write( "the matrix across the bottom row since it contains a 0:\r\n" );
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document.write( "We find the inverse of A by the Gauss-Jordan method.\r\n" );
document.write( "We augment A with the identity matrix:\r\n" );
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document.write( "Use row operations to get the identity on the left:\r\n" );
document.write( "Swap the 1st and 3rd rows:\r\n" );
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document.write( "Multiply R1 by -1 and add to R2, restoring R1 after multiplying:\r\n" );
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document.write( "Multiply R1 by -3 and add to R3, restoring R1 after multiplying:\r\n" );
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document.write( "Multiply R2 by 2 and add to R3, restoring R2 after multiplying:\r\n" );
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document.write( "To avoid fractions till the last step, multiply R1 by 6\r\n" );
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document.write( "Add R3 to R1\r\n" );
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document.write( "Also to avoid fractions till the last step, multiply\r\n" );
document.write( "R2 by -3\r\n" );
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document.write( "Add R3 to R2\r\n" );
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document.write( "Divide R1 through by 6,\r\n" );
document.write( "Divide R2 through by -3\r\n" );
document.write( "Divide R3 through by -6\r\n" );
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document.write( "So the inverse of A is the right half of the above,\r\n" );
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document.write( "Now find the adjoint of\r\n" );
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document.write( "First we find the matrix of signed minor determinants:\r\n" );
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document.write( "which is\r\n" );
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document.write( "Then its transpose is the adjoint:\r\n" );
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document.write( "\"adj%28A%29=%28matrix%283%2C3%2C1%2C2%2C1%2C-2%2C2%2C4%2C-1%2C-2%2C5%29%29\"\r\n" );
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document.write( "Then we substitute in the formula\r\n" );
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document.write( "\"matrix%281%2C3%2C%0D%0A%0D%0AA%5E%28-1%29%2C%22%22=%22%22%2Cadj%28A%29%2Fabs%28A%29%29\"\r\n" );
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document.write( "which says to divide every element of the adjoint matrix\r\n" );
document.write( "by the determinant of A, which we have calculated to be 6.\r\n" );
document.write( "So we get the same inverse:\r\n" );
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document.write( "Both methods are tedious and very time-consuming by hand. You are prone to make\r\n" );
document.write( "a mistake.  Since we now have graphing calculators which can find inverses\r\n" );
document.write( "easily, that's the way we should do them.\r\n" );
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document.write( "Next we left multiply both sides of this matrix equation:\r\n" );
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document.write( "by the inverse matrix\r\n" );
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document.write( "and get this equation:\r\n" );
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document.write( "Now I will assume that you know how to multiply two matrices.  If\r\n" );
document.write( "you don't know how, then post again asking how to.  When you\r\n" );
document.write( "multiply the red matrices by the black matrices just to the right of\r\n" );
document.write( "each red ones, you get this:\r\n" );
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document.write( "Notice that when you multiply the inverse of a matrix by the matrix\r\n" );
document.write( "of which it is the inverse you get the identity matrix, which has\r\n" );
document.write( "1's on the diagonal and 0's elsewhere.\r\n" );
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document.write( "Now if you multiply the two matrices on the left and on the right\r\n" );
document.write( "and you get:\r\n" );
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document.write( "\"%28matrix%283%2C1%2Cx%2Cy%2Cz%29%29\"\"%22%22=%22%22\"\"%28matrix%283%2C1%2C3%2F2%2C0%2C-1%2F2%29%29\"\r\n" );
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document.write( "and so x=3/2, y=0, and z=-1/2.\r\n" );
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document.write( "Edwin

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