Inverse of [[1,2],[3,4]]: animated lesson

1. 1. The problem
   Worked example

   $\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}^{-1}$
2. 2. Step by step
   1. $$\det A = -2 \neq 0$$
      Check that the determinant is non-zero
   2. $$A^{-1} = \frac{1}{-2}\begin{bmatrix}4 & -2 \\ -3 & 1\end{bmatrix} = \begin{bmatrix}-2 & 1 \\ \frac{3}{2} & -\frac{1}{2}\end{bmatrix}$$
      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. 3. The answer
   $$A^{-1} = \begin{bmatrix}-2 & 1 \\ \frac{3}{2} & -\frac{1}{2}\end{bmatrix}$$

   A⁻¹ = [[-2, 1], [3/2, -1/2]]

4. 4. Check
   • ✓Multiplying A by the result gives the identity matrix exactly.

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

• Video lesson

  Transpose of [[1,2],[3,4],[5,6]]

  ▶0:25
  Linear algebraCollege3 steps

  Transpose of [[1,2],[3,4],[5,6]]

  Matrices

• Video lesson

  $\det \begin{bmatrix}-5 & -5 & 0 \\ -9 & 8 & 7 \\ 8 & -1 & -5\end{bmatrix}$

  ▶0:36
  Linear algebraCollege4 steps

  Find the determinant of [[-5,-5,0],[-9,8,7],[8,-1,-5]]

  Matrices

• Video lesson

  $\det \begin{bmatrix}5 & 7 \\ 2 & 3\end{bmatrix}$

  ▶0:26
  Linear algebraCollege2 steps

  Find the determinant of [[5,7],[2,3]]

  Matrices

• Video lesson

  $\begin{bmatrix}2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4\end{bmatrix}^{-1}$

  ▶0:42
  Linear algebraCollege7 steps

  Inverse of [[2,0,0],[0,3,0],[0,0,4]]

  Matrices

• Video lesson

  Find the determinant of [[1,2,0,1],[0,1,3,2],[2,0,1,1],[1,1,1,1]]

  ▶0:53
  Linear algebraCollege6 steps

  Find the determinant of [[1,2,0,1],[0,1,3,2],[2,0,1,1],[1,1,1,1]]

  Matrices

• Video lesson

  $\det \begin{bmatrix}4 & 8 & 8 \\ 3 & 2 & 9 \\ 3 & -1 & -6\end{bmatrix}$

  ▶0:35
  Linear algebraCollege4 steps

  Find the determinant of [[4,8,8],[3,2,9],[3,-1,-6]]

  Matrices

• All matrices lessons
• Browse the whole lesson library
• Make a lesson from your own problem
• Step-by-step solver: check your own answer.