Finding the Inverse of a 2x2 Matrix from Scratch

Finding the Inverse of a 2x2 Matrix from Scratch

This post shows a complete, step-by-step derivation of the inverse of a 2x2 matrix. Everything is expressed using stable, browser-safe ASCII formatting so the layout displays correctly on all devices and all templates.

FIRST PART.

Start with the matrix equation:

A = [[a, b], [c, d]]
A^(-1) = [[w, x], [y, z]]

Goal:  A * A^(-1) = I

This produces the column equations:

[aw + by, cw + dy]^T = [1, 0]^T
[ax + bz, cx + dz]^T = [0, 1]^T

Which gives the four equations:

aw + by = 1
cw + dy = 0
ax + bz = 0
cx + dz = 1

SECOND PART.

Use the first two equations to find w.

aw + by = 1
cw + dy = 0

Multiply:

(ad)w + (bd)y = d      (first eq multiplied by d)
(bc)w + (bd)y = 0      (second eq multiplied by b)

Subtract:

(ad - bc)w = d

w = d / (ad - bc)
(ad - bc != 0)

THIRD PART.

Use the next pair to find x.

ax + bz = 0
cx + dz = 1

Multiply:

(ad)x + (bd)z = 0
(bc)x + (bd)z = b

Subtract:

(ad - bc)x = -b

x = -b / (ad - bc)
(ad - bc != 0)

FOURTH PART.

Find y from cw + dy = 0.

cw + dy = 0
dy = -cw
y = (-cw) / d

Substitute w = d / (ad - bc):

y = (-c / d) * (d / (ad - bc))
y = -c / (ad - bc)
(ad - bc != 0)

FIFTH PART.

Find z from ax + bz = 0.

ax + bz = 0
bz = -ax

Substitute x = -b / (ad - bc):

bz = (-a) * (-b / (ad - bc))
bz = ab / (ad - bc)

z = a / (ad - bc)
(ad - bc != 0)

FINAL PART.

Collect the four entries:

w =  d / (ad - bc)
x = -b / (ad - bc)
y = -c / (ad - bc)
z =  a / (ad - bc)

So the matrix inverse is:

A          = [[a, b], [c, d]]
A^(-1)     = (1 / (ad - bc)) * [[d, -b], [-c, a]]
Condition  = ad - bc != 0

Check:

A * A^(-1) = I
A^(-1) * A = I

And:

det(A) = ad - bc

FINAL NOTES.

  • If det(A) = 0, then A is singular and has no inverse.
  • The same idea applies to 3x3 matrices: determinant zero means no inverse.
  • The minor of an element in a 3x3 matrix is the determinant of the submatrix left after removing the element’s row and column.

Personalised notes based on FP3, Edexcel, Pearson.

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