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Inverse of [[2,1],[1,1]]: animated lesson

A narrated, step-by-step matrices video lesson with captions. Final answer: A⁻¹ = [[1, -1], [-1, 2]].

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.2 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\begin{bmatrix}2 & 1 \\ 1 & 1\end{bmatrix}^{-1}$
2. Step by step
1. $\det A = 1 \neq 0$
  Check that the determinant is non-zero
2. $A^{-1} = \frac{1}{1}\begin{bmatrix}1 & -1 \\ -1 & 2\end{bmatrix} = \begin{bmatrix}1 & -1 \\ -1 & 2\end{bmatrix}$
  Swap a and d, negate b and c, divide by the determinant
3. $A A^{-1} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$
  Check: A·A⁻¹ = I
3. The answer
$A^{-1} = \begin{bmatrix}1 & -1 \\ -1 & 2\end{bmatrix}$
A⁻¹ = [[1, -1], [-1, 2]]
4. Check
- Multiplying A by the result gives the identity matrix exactly.

Transcript

The problem. Here is the problem: [[2, 1], [1, 1]]. Find the inverse of a 2 times 2 matrix.

Step by step. Check that the determinant is non-zero. Swap a and d, negate b and c, divide by the determinant. Check: A times A to the power minus 1 equals I.

The answer. So the answer is A to the power minus 1 equals [[1, negative 1], [-1, 2]]. The determinant is 1, so A is invertible. Every entry of the inverse is exact.

Check. Multiplying A by the result gives the identity matrix exactly.

About matrices

Determinants, inverses, products, transposes and eigenvalues of matrices.

Inverse of [[2,1],[1,1]]: Matrices Step-by-Step Lesson