The inverse of a matrix

To solve 3x = 9 we multiply by the inverse of 3. Can the same be done with A X = B, writing X = A⁻¹ B? Yes — but only when A has an inverse, which is not always the case.

Definition

A square matrix A is invertible if there is a matrix A⁻¹ such that:

A × A⁻¹ = A⁻¹ × A = I

A⁻¹ is called the inverse of A. It is unique when it exists. A non-invertible matrix is called singular.

Two remarks that prevent mistakes: an inverse only exists for square matrices, and you must never write B/A — matrix division does not exist, only multiplication by the inverse makes sense, and on the right side.

The 2 × 2 case: a formula worth knowing

        [ a  b ]                    1     [  d  -b ]
  A =   [ c  d ]     ->    A⁻¹ = ------- [ -c   a ]
                                 ad - bc

Swap the diagonal entries, flip the sign of the others, and divide by ad - bc. That number is the determinant of A. Everything hinges on it: if it is zero, the division is impossible and the matrix is not invertible.

An example:

A = [ 3  1 ]     det = 3×2 - 1×5 = 1
    [ 5  2 ]

A⁻¹ = [  2  -1 ]      check:  [ 3  1 ] [  2  -1 ]   [ 1  0 ]
      [ -5   3 ]              [ 5  2 ]×[ -5   3 ] = [ 0  1 ]  ✔

Solving a system with the inverse

If A is invertible, the system A X = B has a unique solution:

   A X = B
A⁻¹A X = A⁻¹B
     X = A⁻¹B

On the previous example, with B = (9 ; 16):

X = [  2  -1 ] [  9 ]   [ 2×9 - 1×16 ]   [ 2 ]
    [ -5   3 ]×[ 16 ] = [ -5×9 + 3×16 ] = [ 3 ]

so x = 2 and y = 3.

Mind the side. Since the product is not commutative, you multiply on the left on both sides: from A X = B you get X = A⁻¹B, definitely not B A⁻¹ — which, besides being wrong, does not even have compatible dimensions.

What the inverse does to operations

(A⁻¹)⁻¹ = A                        inverting twice changes nothing
(AB)⁻¹  = B⁻¹ A⁻¹                  the order is REVERSED
(tA)⁻¹  = t(A⁻¹)

The second rule makes sense through transformations: if you put on socks then shoes, you must take off the shoes before the socks. Undoing a composition means undoing each step in reverse order.

A matrix with no inverse

A = [ 1  2 ]     det = 1×4 - 2×2 = 0
    [ 2  4 ]     (the 2nd column is twice the 1st: redundant information)

No matrix can invert A: the associated map crushes the plane onto a line, and that collapse is irreversible.

Summary

• A is invertible if A A⁻¹ = A⁻¹ A = I; the inverse is then unique.
• Only square matrices can be invertible; matrix division does not exist.
• For 2 × 2: swap the diagonal, flip the signs of the others, divide by ad - bc.
• det(A) = 0 → singular matrix, not invertible.
• If A is invertible, A X = B has the unique solution X = A⁻¹B, multiplying on the left.
• (AB)⁻¹ = B⁻¹A⁻¹: the inverse of a product reverses the order.