SOLUTION: find inverse: *both matrices are 2x2. Rows separated by brackets* [[6,-3][7,9]]X=[[-18,-93][4,-21]] Please help. I have no idea how to do this.

Algebra ->  Matrices-and-determiminant -> SOLUTION: find inverse: *both matrices are 2x2. Rows separated by brackets* [[6,-3][7,9]]X=[[-18,-93][4,-21]] Please help. I have no idea how to do this.       Log On


   

Question 982591: find inverse:
*both matrices are 2x2. Rows separated by brackets*
[[6,-3][7,9]]X=[[-18,-93][4,-21]]
Please help. I have no idea how to do this.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Find the inverse of the matrix %28matrix%282%2C2%2C6%2C-3%2C7%2C9%29%29

Solved by pluggable solver: Finding the Inverse of a 2x2 Matrix

To find the inverse of the matrix A=%28matrix%282%2C2%2C6%2C-3%2C7%2C9%29%29, we can follow these steps:

Step 1) Find the determinant



The determinant of %28matrix%282%2C2%2C6%2C-3%2C7%2C9%29%29 is abs%28matrix%282%2C2%2C6%2C-3%2C7%2C9%29%29=75. So this means that d=75

Step 2) Swap the values



Now switch the highlighted values %28matrix%282%2C2%2Chighlight%286%29%2C-3%2C7%2Chighlight%289%29%29%29 to get %28matrix%282%2C2%2Chighlight%289%29%2C-3%2C7%2Chighlight%286%29%29%29

Step 3) Change the sign



Now change the sign of the highlighted values %28matrix%282%2C2%2C9%2Chighlight%28-3%29%2Chighlight%287%29%2C6%29%29 to get %28matrix%282%2C2%2C9%2Chighlight%283%29%2Chighlight%28-7%29%2C6%29%29

Step 4) Multiply by the inverse of the determinant



Multiply by 1%2Fd to get %281%2Fd%29%28matrix%282%2C2%2C9%2C3%2C-7%2C6%29%29

Plug in d=75 to get %281%2F75%29%28matrix%282%2C2%2C9%2C3%2C-7%2C6%29%29

Step 5) Multiply 1%2F75 by every element in the matrix (simplify and reduce if possible)



Multiply 1%2F75 by EVERY element to get

Multiply to get %28matrix%282%2C2%2C9%2F75%2C3%2F75%2C-7%2F75%2C6%2F75%29%29

Reduce each element: %28matrix%282%2C2%2C3%2F25%2C1%2F25%2C-7%2F75%2C2%2F25%29%29


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Answer:

So the inverse of %28matrix%282%2C2%2C6%2C-3%2C7%2C9%29%29 is %28matrix%282%2C2%2C3%2F25%2C1%2F25%2C-7%2F75%2C2%2F25%29%29

This means that if A=%28matrix%282%2C2%2C6%2C-3%2C7%2C9%29%29 then A%5E%28-1%29=%28matrix%282%2C2%2C3%2F25%2C1%2F25%2C-7%2F75%2C2%2F25%29%29



Now multiply both sides by the inverse matrix. The left side matrices will multiply to the identity matrix, leaving just matrix X behind.

On the right side, you will have to compute the matrix product to get the final answer for X