Let's solve the given matrix equation for X.
The equation is:
[−2−7−6−6]X+[4−3−7−8]=[−9−81−5]
First, isolate X:
[−2−7−6−6]X=[−9−81−5]−[4−3−7−8]
Calculate the matrix on the right side:
[−9−81−5]−[4−3−7−8]=[−9−4−8+31+7−5+8]=[−13−583]
So the equation becomes:
[−2−7−6−6]X=[−13−583]
To solve for X, multiply both sides by the inverse of the matrix [−2−7−6−6].
First, find the inverse of [−2−7−6−6]:
The inverse of a 2x2 matrix [acbd] is ad−bc1[d−c−ba].
For [−2−7−6−6]:
Determinant=(−2×−6)−(−6×−7)=12−42=−30
The inverse is:
−301[−676−2]=[51−307−51151]
Now, multiply the inverse by the matrix [−13−583]:
X=[51−307−51151][−13−583]
Calculate each element of the resulting matrix:
-
First element:
X11=(51×−13)+(−51×−5)=−513+55=−513+1=−513−5=−58=−58
-
Second element:
X12=(51×8)+(−51×3)=58−53=58−3=55=1
-
Third element:
X21=(−307×−13)+(151×−5)=3091−155=3091−3010=3081=1027=2.7
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Fourth element:
X22=(−307×8)+(151×3)=−3056+153=−3056+306=−3050=−35
Therefore, the solution is:
X=[−582.71−35]
Would you like more details or have any questions?
Here are 5 questions you might want to ask:
- How can I verify the correctness of a matrix inverse?
- Can you explain the determinant calculation in more detail?
- How does matrix multiplication work?
- What are some common mistakes when solving matrix equations?
- Can you show me how to solve another matrix equation?
Tip: Always double-check your calculations, especially when working with fractions and negative signs, to avoid common arithmetic errors.