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Inverse of a 2×2 Matrix

Matrices · Class XII

When det ≠ 0, the inverse exists: swap diagonals, negate off-diagonals, divide by det.

1
type a value
-55
2
type a value
-55
3
type a value
-55
4
type a value
-55
Live values
  • det-2
  • Inverse [1,1]-2
  • Inverse [1,2]1
  • Inverse [2,1]1.5
  • Inverse [2,2]-0.5
xy
  • Inverse[1,1] = d/(a*d - b*c) vs d

Formulas in this lab

  • det
    det⁡=ad−bc\det = ad - bc
  • Inverse [1,1]
    d/det⁡d/\det
  • Inverse [1,2]
    −b/det⁡-b/\det
  • Inverse [2,1]
    −c/det⁡-c/\det
  • Inverse [2,2]
    a/det⁡a/\det
Tip: Watch the entry blow up at d = b*c/a , that is exactly where det = 0 and the inverse fails.

Frequently asked questions

▶What is the inverse of a $2\times 2$ matrix?

For $M = \begin{pmatrix}a & b\\ c & d\end{pmatrix}$ with $\det M = ad - bc \ne 0$, the inverse is $M^{-1} = \frac{1}{ad - bc}\begin{pmatrix}d & -b\\ -c & a\end{pmatrix}$. It satisfies $MM^{-1} = I$.

▶How do I use the $2\times 2$ inverse lab?

Slide the four entries and the lab returns $\det M$ and all four entries of $M^{-1}$. The lab also shows $M\cdot M^{-1}$, which should always equal the identity. Try $a = 2$, $b = 1$, $c = 1$, $d = 1$: $\det = 1$ and $M^{-1} = \begin{pmatrix}1 & -1\\ -1 & 2\end{pmatrix}$.

▶Why must $\det M \ne 0$ for the inverse to exist?

The inverse formula divides by $\det M$. If $\det M = 0$, the matrix is singular and represents a system with either no solution or infinitely many. Students sometimes plug zero anyway and get NaN — the lab catches this for you.

▶Where is matrix inversion used in real applications?

Solving systems of linear equations $Mx = b$ via $x = M^{-1}b$ underlies GPS positioning, regression in data science, and balancing chemical equations. Engineers solving structural load equations or economists fitting demand models all invert matrices.