document.write( "Question 355472: use matrix inversion to solve 3x-4y=-1 -2x+3y=3 \n" ); document.write( "
Algebra.Com's Answer #253882 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"system%283x-4y=-1%2C++%0D%0A-2x%2B3y=3%29\"\r\n" );
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document.write( "Write that system in the matrix form AX = B\r\n" );
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document.write( "First find the inverse of \"%28matrix%282%2C2%2C3%2C-4%2C-2%2C3%29%29\":\r\n" );
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document.write( "To do that:\r\n" );
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document.write( "1.  Find the value of its determinant, \"abs%28matrix%282%2C2%2C3%2C-4%2C-2%2C3%29%29=%283%29%283%29-%28-4%29%28-2%29+=+9-8+=+1\"\r\n" );
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document.write( "2. Swap the upper left and lower right elements of \"%28matrix%282%2C2%2C3%2C-4%2C-2%2C3%29%29\",\r\n" );
document.write( "getting \"%28matrix%282%2C2%2C3%2C-4%2C-2%2C3%29%29\"\r\n" );
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document.write( "[In this case it didn't change anything since they were both 3, but in\r\n" );
document.write( "other problems it will be different and you must swap them]\r\n" );
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document.write( "3. Then change the signs of the upper right and lower left elements,\r\n" );
document.write( "getting \"%28matrix%282%2C2%2C3%2C4%2C2%2C3%29%29\"\r\n" );
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document.write( "4. Divide every element by the value of the determinant of the \r\n" );
document.write( "original matrix which we found to be 13 in step 1, getting\r\n" );
document.write( "\"%28matrix%282%2C2%2C3%2F1%2C4%2F1%2C2%2F1%2C3%2F1%29%29\" or \"%28matrix%282%2C2%2C3%2C4%2C2%2C3%29%29\"\r\n" );
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document.write( "[In this case it didn't change anything since the determinant was 1, but\r\n" );
document.write( "in other problems it will be different and you must divide.]\r\n" );
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document.write( "This is the inverse A-1 of\r\n" );
document.write( "the original matrix A.\r\n" );
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document.write( "Left-multiply both sides of the given matrix equation, AX=B\r\n" );
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document.write( "by this inverse, getting the form A-1(AX)=A-1B\r\n" );
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document.write( "Use the associative principle to move the parentheses around\r\n" );
document.write( "the first two matrices on the left, getting the form (A-1A)X=A-1B\r\n" );
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document.write( "Do the matrix multiplication:\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "Simplify some more:\r\n" );
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document.write( "This is the form IX=A-1B\r\n" );
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document.write( "Multiply the matrices on the left:\r\n" );
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document.write( "\"%28matrix%282%2C1%2C1%2Ax%2B0%2Ay%2C0%2Ax%2B1%2Ay%29%29=%28matrix%282%2C1%2C9%2C7%29%29\"\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "\"%28matrix%282%2C1%2Cx%2Cy%29%29=%28matrix%282%2C1%2C9%2C7%29%29\".  This is the form X=A-1B.\r\n" );
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document.write( "So the solution is \"x=9\" and \"y=7\"\r\n" );
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document.write( "Edwin
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