document.write( "Question 627192: Find the inverse of the matrix if it exists.?
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Algebra.Com's Answer #394750 by jim_thompson5910(35256)\"\" \"About 
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Solved by pluggable solver: Finding the Inverse of a 2x2 Matrix

To find the inverse of the matrix \"A=%28matrix%282%2C2%2C1%2C-2%2C1%2C2%29%29\", we can follow these steps:

Step 1) Find the determinant



The determinant of \"%28matrix%282%2C2%2C1%2C-2%2C1%2C2%29%29\" is \"abs%28matrix%282%2C2%2C1%2C-2%2C1%2C2%29%29=4\". So this means that \"d=4\"

Step 2) Swap the values



Now switch the highlighted values \"%28matrix%282%2C2%2Chighlight%281%29%2C-2%2C1%2Chighlight%282%29%29%29\" to get \"%28matrix%282%2C2%2Chighlight%282%29%2C-2%2C1%2Chighlight%281%29%29%29\"

Step 3) Change the sign



Now change the sign of the highlighted values \"%28matrix%282%2C2%2C2%2Chighlight%28-2%29%2Chighlight%281%29%2C1%29%29\" to get \"%28matrix%282%2C2%2C2%2Chighlight%282%29%2Chighlight%28-1%29%2C1%29%29\"

Step 4) Multiply by the inverse of the determinant



Multiply by \"1%2Fd\" to get \"%281%2Fd%29%28matrix%282%2C2%2C2%2C2%2C-1%2C1%29%29\"

Plug in \"d=4\" to get \"%281%2F4%29%28matrix%282%2C2%2C2%2C2%2C-1%2C1%29%29\"

Step 5) Multiply \"1%2F4\" by every element in the matrix (simplify and reduce if possible)



Multiply \"1%2F4\" by EVERY element to get

Multiply to get \"%28matrix%282%2C2%2C2%2F4%2C2%2F4%2C-1%2F4%2C1%2F4%29%29\"

Reduce each element: \"%28matrix%282%2C2%2C1%2F2%2C1%2F2%2C-1%2F4%2C1%2F4%29%29\"


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Answer:

So the inverse of \"%28matrix%282%2C2%2C1%2C-2%2C1%2C2%29%29\" is \"%28matrix%282%2C2%2C1%2F2%2C1%2F2%2C-1%2F4%2C1%2F4%29%29\"

This means that if \"A=%28matrix%282%2C2%2C1%2C-2%2C1%2C2%29%29\" then \"A%5E%28-1%29=%28matrix%282%2C2%2C1%2F2%2C1%2F2%2C-1%2F4%2C1%2F4%29%29\"
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