SOLUTION: How do you solve this matrix equation? [-7 -9] X [3 4] = [1 9] [4 5] + [4 -3] = [6 -6]

Algebra ->  Matrices-and-determiminant -> SOLUTION: How do you solve this matrix equation? [-7 -9] X [3 4] = [1 9] [4 5] + [4 -3] = [6 -6]      Log On


   

Question 176026This question is from textbook Algebra 2
: How do you solve this matrix equation?
[-7 -9] X [3 4] = [1 9]
[4 5] + [4 -3] = [6 -6] This question is from textbook Algebra 2

Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
How do you solve this matrix equation?
[-7 -9] X [3 4] = [1 9]
[4 5] + [4 -3] = [6 -6]
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Convert to two equations:
-7*3 + -9*4 = 1
That is wrong so something is wrong with your posting.
Cheers,
Stan H.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Start with the given equation.


Subtract %28matrix%282%2C2%2C3%2C4%2C4%2C-3%29%29 from both sides.


%28matrix%282%2C2%2C-7%2C-9%2C4%2C5%29%29X=%28matrix%282%2C2%2C-2%2C5%2C2%2C-3%29%29 Subtract (note: to subtract the matrices, simply subtract the corresponding components)


Now let's find the inverse of %28matrix%282%2C2%2C-7%2C-9%2C4%2C5%29%29

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

To find the inverse of the matrix A=%28matrix%282%2C2%2C-7%2C-9%2C4%2C5%29%29, we can follow these steps:

Step 1) Find the determinant



The determinant of %28matrix%282%2C2%2C-7%2C-9%2C4%2C5%29%29 is abs%28matrix%282%2C2%2C-7%2C-9%2C4%2C5%29%29=1. So this means that d=1

Step 2) Swap the values



Now switch the highlighted values %28matrix%282%2C2%2Chighlight%28-7%29%2C-9%2C4%2Chighlight%285%29%29%29 to get %28matrix%282%2C2%2Chighlight%285%29%2C-9%2C4%2Chighlight%28-7%29%29%29

Step 3) Change the sign



Now change the sign of the highlighted values %28matrix%282%2C2%2C5%2Chighlight%28-9%29%2Chighlight%284%29%2C-7%29%29 to get %28matrix%282%2C2%2C5%2Chighlight%289%29%2Chighlight%28-4%29%2C-7%29%29

Step 4) Multiply by the inverse of the determinant



Multiply by 1%2Fd to get %281%2Fd%29%28matrix%282%2C2%2C5%2C9%2C-4%2C-7%29%29

Plug in d=1 to get %281%29%28matrix%282%2C2%2C5%2C9%2C-4%2C-7%29%29

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



Multiply 1 by EVERY element to get

Multiply to get %28matrix%282%2C2%2C5%2F1%2C9%2F1%2C-4%2F1%2C-7%2F1%29%29

Reduce each element: %28matrix%282%2C2%2C5%2C9%2C-4%2C-7%29%29


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Answer:

So the inverse of %28matrix%282%2C2%2C-7%2C-9%2C4%2C5%29%29 is %28matrix%282%2C2%2C5%2C9%2C-4%2C-7%29%29

This means that if A=%28matrix%282%2C2%2C-7%2C-9%2C4%2C5%29%29 then A%5E%28-1%29=%28matrix%282%2C2%2C5%2C9%2C-4%2C-7%29%29




-------------------------------------------------------------


%28matrix%282%2C2%2C-7%2C-9%2C4%2C5%29%29X=%28matrix%282%2C2%2C-2%2C5%2C2%2C-3%29%29 Go back to the equation


Left multiply both sides by the inverse %28matrix%282%2C2%2C5%2C9%2C-4%2C-7%29%29



Multiply %28matrix%282%2C2%2C5%2C9%2C-4%2C-7%29%29 and %28matrix%282%2C2%2C-7%2C-9%2C4%2C5%29%29 to get %28matrix%282%2C2%2C1%2C0%2C0%2C1%29%29



%28matrix%282%2C2%2C1%2C0%2C0%2C1%29%29X=%28matrix%282%2C2%2C8%2C-2%2C-6%2C1%29%29 Multiply %28matrix%282%2C2%2C5%2C9%2C-4%2C-7%29%29 and %28matrix%282%2C2%2C-2%2C5%2C2%2C-3%29%29 to get %28matrix%282%2C2%2C8%2C-2%2C-6%2C1%29%29



X=%28matrix%282%2C2%2C8%2C-2%2C-6%2C1%29%29 Multiply %28matrix%282%2C2%2C1%2C0%2C0%2C1%29%29 (which is I) and X to get I%2AX=X



So the answer is


X=%28matrix%282%2C2%2C8%2C-2%2C-6%2C1%29%29