Inverse matrix
The inverse of A, written A^(-1), satisfies A times A^(-1) = I. For [[a,b],[c,d]] it is (1/(ad - bc)) times [[d,-b],[-c,a]].
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Analogy
An undo button: the inverse reverses whatever the matrix did.
Example
det = 6 - 5 = 1.Swap diagonal, negate others.Inverse = [[3,-1],[-5,2]].Check: row 1 dot col 1 = 6 - 5 = 1.
In practice
Lets you solve equations by multiplying, rather than eliminating.
Practice
Flashcards
- What does a zero determinant mean?The matrix has no inverse.
- What is the inverse of [[2,1],[5,3]]?[[3,-1],[-5,2]].
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
When does a matrix have an inverse?
Exactly when it is square and its determinant is not zero. Such a matrix is called invertible or non-singular, and its columns are independent. If the determinant is zero, as for [[1,2],[2,4]], the matrix squashes space and nothing can undo it.
How do I check my inverse is correct?
Multiply the matrix by the inverse you found. You should get the identity matrix, with ones on the diagonal and zeros elsewhere. For [[2,1],[5,3]] and [[3,-1],[-5,2]], the product is [[1,0],[0,1]]. Do not skip this quick check, because one wrong sign is easy to make.