document.write( "Question 365778: 7 2 determine whether the matrix has an inverse. If an inverse exists, find it.
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Algebra.Com's Answer #260706 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
A matrix has an inverse if the determinant is nonzero.\r
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Solved by pluggable solver: Finding the Determinant of a 2x2 Matrix

If you have the general 2x2 matrix:

\"%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\"

the determinant is: \"D=a%2Ad-c%2Ab\"

So this means that

\"abs%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29=a%2Ad-c%2Ab\"

Note: the vertical bars denote a determinant.


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So in this case the determinant of \"%28matrix%282%2C2%2C7%2C2%2C0%2C-3%29%29\" is:

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\"abs%28matrix%282%2C2%2C7%2C2%2C0%2C-3%29%29=%287%29%28-3%29-%280%29%282%29=-21-0=-21\"


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Answer:

So \"abs%28matrix%282%2C2%2C7%2C2%2C0%2C-3%29%29=-21\" which means that the determinant of the matrix \"%28matrix%282%2C2%2C7%2C2%2C0%2C-3%29%29\" is -21

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\n" ); document.write( "\n" ); document.write( "Since the determinant is nonzero, this means that the inverse exists.\r
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\n" ); document.write( "\n" ); document.write( "So let's find the inverse...\r
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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%2C7%2C2%2C0%2C-3%29%29\", we can follow these steps:

Step 1) Find the determinant



The determinant of \"%28matrix%282%2C2%2C7%2C2%2C0%2C-3%29%29\" is \"abs%28matrix%282%2C2%2C7%2C2%2C0%2C-3%29%29=-21\". So this means that \"d=-21\"

Step 2) Swap the values



Now switch the highlighted values \"%28matrix%282%2C2%2Chighlight%287%29%2C2%2C0%2Chighlight%28-3%29%29%29\" to get \"%28matrix%282%2C2%2Chighlight%28-3%29%2C2%2C0%2Chighlight%287%29%29%29\"

Step 3) Change the sign



Now change the sign of the highlighted values \"%28matrix%282%2C2%2C-3%2Chighlight%282%29%2Chighlight%280%29%2C7%29%29\" to get \"%28matrix%282%2C2%2C-3%2Chighlight%28-2%29%2Chighlight%280%29%2C7%29%29\"

Step 4) Multiply by the inverse of the determinant



Multiply by \"1%2Fd\" to get \"%281%2Fd%29%28matrix%282%2C2%2C-3%2C-2%2C0%2C7%29%29\"

Plug in \"d=-21\" to get \"%28-1%2F21%29%28matrix%282%2C2%2C-3%2C-2%2C0%2C7%29%29\"

Step 5) Multiply \"-1%2F21\" by every element in the matrix (simplify and reduce if possible)



Multiply \"-1%2F21\" by EVERY element to get

Multiply to get \"%28matrix%282%2C2%2C-3%2F-21%2C-2%2F-21%2C0%2F-21%2C7%2F-21%29%29\"

Reduce each element: \"%28matrix%282%2C2%2C1%2F7%2C2%2F21%2C0%2C-1%2F3%29%29\"


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Answer:

So the inverse of \"%28matrix%282%2C2%2C7%2C2%2C0%2C-3%29%29\" is \"%28matrix%282%2C2%2C1%2F7%2C2%2F21%2C0%2C-1%2F3%29%29\"

This means that if \"A=%28matrix%282%2C2%2C7%2C2%2C0%2C-3%29%29\" then \"A%5E%28-1%29=%28matrix%282%2C2%2C1%2F7%2C2%2F21%2C0%2C-1%2F3%29%29\"
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