document.write( "Question 176018This question is from textbook Algebra 2
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Algebra.Com's Answer #131136 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
First, we need to find the inverse of the matrix \"%28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29\"\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%2C2%2C4%2C0%2C1%29%29\", we can follow these steps:

Step 1) Find the determinant



The determinant of \"%28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29\" is \"abs%28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29=2\". So this means that \"d=2\"

Step 2) Swap the values



Now switch the highlighted values \"%28matrix%282%2C2%2Chighlight%282%29%2C4%2C0%2Chighlight%281%29%29%29\" to get \"%28matrix%282%2C2%2Chighlight%281%29%2C4%2C0%2Chighlight%282%29%29%29\"

Step 3) Change the sign



Now change the sign of the highlighted values \"%28matrix%282%2C2%2C1%2Chighlight%284%29%2Chighlight%280%29%2C2%29%29\" to get \"%28matrix%282%2C2%2C1%2Chighlight%28-4%29%2Chighlight%280%29%2C2%29%29\"

Step 4) Multiply by the inverse of the determinant



Multiply by \"1%2Fd\" to get \"%281%2Fd%29%28matrix%282%2C2%2C1%2C-4%2C0%2C2%29%29\"

Plug in \"d=2\" to get \"%281%2F2%29%28matrix%282%2C2%2C1%2C-4%2C0%2C2%29%29\"

Step 5) Multiply \"1%2F2\" by every element in the matrix (simplify and reduce if possible)



Multiply \"1%2F2\" by EVERY element to get

Multiply to get \"%28matrix%282%2C2%2C1%2F2%2C-4%2F2%2C0%2F2%2C2%2F2%29%29\"

Reduce each element: \"%28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29\"


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Answer:

So the inverse of \"%28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29\" is \"%28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29\"

This means that if \"A=%28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29\" then \"A%5E%28-1%29=%28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29\"

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\n" ); document.write( "\n" ); document.write( "Now let's use the inverse matrix to find the matrix X:\r
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\n" ); document.write( "\n" ); document.write( "\"%28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29X=%28matrix%282%2C3%2C4%2C0%2C6%2C3%2C-1%2C5%29%29\" Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( " Left multiply both sides by the inverse \"%28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29\"\r
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\n" ); document.write( "\n" ); document.write( " Multiply \"%28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29\" and \"%28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29\" to get \"%28matrix%282%2C2%2C1%2C0%2C0%2C1%29%29\" (this is the identity matrix). Let me know if you need help multiplying matrices.\r
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\n" ); document.write( "\n" ); document.write( "\"%28matrix%282%2C2%2C1%2C0%2C0%2C1%29%29X=%28matrix%282%2C3%2C-4%2C2%2C-7%2C3%2C-1%2C5%29%29\" Multiply \"%28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29\" and \"%28matrix%282%2C3%2C4%2C0%2C6%2C3%2C-1%2C5%29%29\" to get \"%28matrix%282%2C3%2C-4%2C2%2C-7%2C3%2C-1%2C5%29%29\".\r
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\n" ); document.write( "\n" ); document.write( "\"X=%28matrix%282%2C3%2C-4%2C2%2C-7%2C3%2C-1%2C5%29%29\" Multiply the identity matrix (which is I) by X to get \"I%2AX=X\"\r
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\n" ); document.write( "\n" ); document.write( "So the answer is \r
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\n" ); document.write( "\n" ); document.write( "\"X=%28matrix%282%2C3%2C-4%2C2%2C-7%2C3%2C-1%2C5%29%29\" \n" ); document.write( "
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