Solving with the inverse
If A x = b and A is invertible, then x = A^(-1) b. Multiply both sides on the left by the inverse.
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Analogy
Unlocking a door: you apply the reverse of the lock's action.
Example
Inverse = [[3,-1],[-5,2]].x = (3(4) - 11, -5(4) + 2(11)).= (1, 2).Check: 2 + 2 = 4, 5 + 6 = 11.
In practice
Handy when you need to solve many systems with the same A but different right-hand sides.
Practice
Flashcards
- How do you solve A x = b with an inverse?x = A inverse times b.
Interview questions
- Find the inverse of [[2,1],[5,3]] and use it to solve A x = (4, 11).
- Determinant is 6 - 5 = 1
- Inverse is [[3,-1],[-5,2]]
- x = (3(4) - 11, -5(4) + 2(11))
- Solution (1, 2)
- Check in the original equations
Questions people ask
Why multiply on the left by the inverse?
Matrix multiplication depends on order. Starting from A x equals b, you must put the inverse on the left of both sides to get A inverse times A times x, which becomes I x, or just x. Multiplying on the right would not cancel.
Is using the inverse better than elimination?
For small systems it is fine, but for large ones elimination is faster and more accurate, because finding an inverse costs more work. The inverse shines when you need to solve with the same matrix repeatedly for different right-hand sides.