## How to Find Inverse of a Matrix

### What is the inverse of Matrix?

Matrix's inverse is another matrix that gives the multiplicative identity when multiplied by the given Matrix. The inverse of a matrix $$A$$ is $$A^{-1}$$ and $$A.A^{-1} \ = \ A^{-1}.A \ = \ I$$ ($$I$$ is the identity matrix). An invertible matrix is one whose determinant is not zero and for which the inverse matrix can be calculated.

### The Formula for the inverse of a Matrix

Consider $$A \ = \ \begin{bmatrix}a & b\\\ c & d\end{bmatrix}$$ as a $$2 \times 2$$ matrix. The formula gives the inverse Matrix of $$A$$.

$$A^{-1} \ = \ \frac{1}{|A|} \ \begin{bmatrix}d & -b\\\ -c & a\end{bmatrix}$$, $$|A| \ = \ ad \ - \ bc$$

As an example, the inverse of $$\begin{bmatrix}1 & 2\\\ 3 & 4\end{bmatrix}$$ is $$\begin{bmatrix}-2 & 1\\\ {3 \over 2} & -{1 \over 2}\end{bmatrix}$$ and we can see that:

$$A.A^{-1} \ = \ \begin{bmatrix}1 & 2\\\ 3 & 4\end{bmatrix} \ \begin{bmatrix}-2 & 1\\\ {3 \over 2} & -{1 \over 2}\end{bmatrix} \ = \ \begin{bmatrix}1 & 0\\\ 0 & 1\end{bmatrix}$$

$$A^{-1}.A \ = \ \begin{bmatrix}-2 & 1\\\ {3 \over 2} & -{1 \over 2}\end{bmatrix} \ \begin{bmatrix}1 & 2\\\ 3 & 4\end{bmatrix} \ = \ \begin{bmatrix}1 & 0\\\ 0 & 1\end{bmatrix}$$

### Exercises for Finding Inverse of a Matrix

1) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 2 & -3 \\\ 1 & 0 \end{bmatrix}$$

2) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} -3 & 8 \\\ 12 & 4 \end{bmatrix}$$

3) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 8 & 3 \\\ 21 & 9 \end{bmatrix}$$

4) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} -2 & 6 \\\ 3 & -5 \end{bmatrix}$$

5) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 11 & 3 \\\ 15 & 4 \end{bmatrix}$$

6) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 4 & 23 \\\ 2 & 12 \end{bmatrix}$$

7) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 9 & 3 \\\ 17 & 6 \end{bmatrix}$$

8) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} -5 & 8 \\\ -4 & 7 \end{bmatrix}$$

9) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 6 & -16 \\\ 2 & -5 \end{bmatrix}$$

10) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 33 & 7 \\\ 9 & 2 \end{bmatrix}$$

1) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 2 & -3 \\\ 1 & 0 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 3}$$
$$\color{red}{A^{-1} \ = \ \frac{1}{3} \ \begin{bmatrix} 0 & 3 \\\ -1 & 2 \end{bmatrix} \ = \ \begin{bmatrix} 0 & 1 \\\ -\frac{1}{3} & \frac{2}{3} \end{bmatrix}}$$

2) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} -3 & 8 \\\ 12 & 4 \end{bmatrix}$$

$$\color{red}{|A| \ = \ -84}$$
$$\color{red}{A^{-1} \ = \ \frac{1}{-84} \ \begin{bmatrix} 4 & -8 \\\ -12 & 3 \end{bmatrix} \ = \ \begin{bmatrix} -\frac{1}{21} & \frac{1}{21} \\\ \frac{1}{7} & -\frac{1}{28} \end{bmatrix}}$$

3) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 8 & 3 \\\ 21 & 9 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 9}$$
$$\color{red}{A^{-1} \ = \ \frac{1}{9} \ \begin{bmatrix} 9 & -3 \\\ -21 & 8 \end{bmatrix} \ = \ \begin{bmatrix} 1 & -\frac{1}{3} \\\ -\frac{7}{3} & \frac{8}{9} \end{bmatrix}}$$

4) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} -2 & 6 \\\ 3 & -5 \end{bmatrix}$$

$$\color{red}{|A| \ = \ -8}$$
$$\color{red}{A^{-1} \ = \ \frac{1}{-8} \ \begin{bmatrix} -5 & -6 \\\ -3 & -2 \end{bmatrix} \ = \ \begin{bmatrix} \frac{5}{8} & \frac{3}{4} \\\ \frac{3}{8} & \frac{1}{4} \end{bmatrix}}$$

5) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 11 & 3 \\\ 15 & 4 \end{bmatrix}$$

$$\color{red}{|A| \ = \ -1}$$
$$\color{red}{A^{-1} \ = \ \frac{1}{-1} \ \begin{bmatrix} 4 & -3 \\\ -15 & 11 \end{bmatrix} \ = \ \begin{bmatrix} -4 & 3 \\\ 15 & -11 \end{bmatrix}}$$

6) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 4 & 23 \\\ 2 & 12 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 2}$$
$$\color{red}{A^{-1} \ = \ \frac{1}{2} \ \begin{bmatrix} 12 & -23 \\\ -2 & 4 \end{bmatrix} \ = \ \begin{bmatrix} 6 & -\frac{23}{2} \\\ -1 & 2 \end{bmatrix}}$$

7) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 9 & 3 \\\ 17 & 6 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 3}$$
$$\color{red}{A^{3} \ = \ \frac{1}{3} \ \begin{bmatrix} 6 & -3 \\\ -17 & 9 \end{bmatrix} \ = \ \begin{bmatrix} 2 & -1 \\\ -\frac{17}{3} & 3 \end{bmatrix}}$$

8) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} -5 & 8 \\\ -4 & 7 \end{bmatrix}$$

$$\color{red}{|A| \ = \ -3}$$
$$\color{red}{A^{-1} \ = \ \frac{1}{-3} \ \begin{bmatrix} 7 & -8 \\\ 4 & -5 \end{bmatrix} \ = \ \begin{bmatrix} -\frac{7}{3} & \frac{8}{3} \\\ \frac{4}{3} & -\frac{7}{3} \end{bmatrix}}$$

9) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 6 & -16 \\\ 2 & -5 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 2}$$
$$\color{red}{A^{-1} \ = \ \frac{1}{2} \ \begin{bmatrix} -5 & 16 \\\ -2 & 6 \end{bmatrix} \ = \ \begin{bmatrix} -\frac{5}{2} & 8 \\\ -1 & 3 \end{bmatrix}}$$

10) Find the inverse of the matrix: $$A \ = \ \begin{bmatrix} 33 & 7 \\\ 9 & 2 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 3}$$
$$\color{red}{A^{-1} \ = \ \frac{1}{3} \ \begin{bmatrix} 2 & -7 \\\ -9 & 33 \end{bmatrix} \ = \ \begin{bmatrix} \frac{2}{3} & -\frac{7}{3} \\\ -3 & 11 \end{bmatrix}}$$

## Finding Inverse of a Matrix Practice Quiz

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