Inverse of a matrix solved example 01. In this post, we will solve one example and provide step by step solution.

## Inverse of a matrix solved example 01

Given matrix:

- Calculate the determinant of the matrix to find the inverse of a matrix:
- Calculate the matrix of minors:
- Calculate the matrix of cofactors (alternating signs):
- Transpose the matrix of cofactors to get the adjugate matrix:
- Calculate the inverse by dividing the adjugate matrix by the determinant:

## Inverse of a matrix solved example 02

Given matrix:

- Calculate the determinant of the matrix:
- Calculate the matrix of minors:
- Calculate the matrix of cofactors (alternating signs):
- Transpose the matrix of cofactors to get the adjugate matrix:
- Calculate the inverse by dividing the adjugate matrix by the determinant:

## Inverse of a matrix solved example

Given matrix:

- Calculate the determinant of the matrix:
- Calculate the matrix of minors:
- Calculate the matrix of cofactors (alternating signs):
- Transpose the matrix of cofactors to get the adjugate matrix:
- Calculate the inverse by dividing the adjugate matrix by the determinant:

## Inverse of a matrix solved example 03

Given matrix:

- Calculate the determinant of the matrix:
- Calculate the matrix of minors:
- Calculate the matrix of cofactors (alternating signs):
- Transpose the matrix of cofactors to get the adjugate matrix:
- Calculate the inverse by dividing the adjugate matrix by the determinant:

## Inverse of a matrix solved example 04

Given matrix:

- Calculate the determinant of the matrix:

- Calculate the matrix of minors:

- Calculate the matrix of cofactors (alternating signs):

- Transpose the matrix of cofactors to get the adjugate matrix:

- Calculate the inverse by dividing the adjugate matrix by the determinant:

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5 & 2 & -3 \\

3 & -1 & 1 \\

-2 & 1 & 2 \\

\end{bmatrix}\text{Det} = 5 \cdot (-1 \cdot 2 – 1 \cdot 1) – 2 \cdot (3 \cdot 2 – 1 \cdot -2) – 3 \cdot (3 \cdot 1 – (-1) \cdot -2) = -2\text{Minors} =

\begin{bmatrix}

1 & -7 & 1 \\

-2 & -8 & -2 \\

-1 & -1 & -5 \\

\end{bmatrix}\text{Cofactors} =

\begin{bmatrix}

1 & 7 & 1 \\

2 & -8 & 2 \\

-1 & 1 & -5 \\

\end{bmatrix}\text{Adjugate} =

\begin{bmatrix}

1 & 2 & -1 \\

7 & -8 & 1 \\

1 & 2 & -5 \\

\end{bmatrix}\text{Inverse} = \frac{1}{\text{Det}} \cdot \text{Adjugate} =

\begin{bmatrix}

-\frac{1}{2} & -1 & \frac{1}{2} \\

-\frac{7}{2} & 4 & -\frac{1}{2} \\

-\frac{1}{2} & -1 & \frac{5}{2} \\

\end{bmatrix}$