Description

In this stage, you should find the inverse of a matrix.

The inverse matrix $A^{−1}$ is the matrix whose product with the original matrix $A$ is equal to the identity matrix.

$A \times A^{-1} = A^{-1} \times A = I$

Watch this video about the inverse of a matrix to get the basic idea. To get a deeper understanding, check out the 3Blue1Brown channel.

The identity matrix is a matrix where all elements of the main diagonal are ones, and other elements are zeros. Here is an example of a $4, 4$ identity matrix:

$I_{4,4} = \begin{pmatrix} 1 & 0 & 0 &0\\ 0 & 1 & 0 &0 \\ 0 & 0 & 1&0 \\0 & 0 & 0 &1\end{pmatrix}$

The inverse of a matrix can be found using this formula:

$A^{-1} = \dfrac{1}{det(A)} \times C^T$

As you can see, it contains a lot of operations you implemented in the previous stages: finding cofactors of all the elements of the matrix, transposition of the matrix, finding the determinant of a matrix, and multiplication of a matrix by a constant.

$det(A)$ is the determinant of matrix $A$, and $C^T$ is the matrix consisting of cofactors of all elements of the matrix $A$ transposed along the main diagonal. The inverse matrix can’t be found if $det(A)$ equals zero. You can look up a calculation example.

Objectives

In this stage, your program should support finding the inverse of a matrix. Refer to the example to see how it should be implemented.

Note that in some cases the inverse of a matrix does not exist. In such cases, your program should output a warning message.

Additional improvements

Although it's not required in this stage and we won't check, you can implement a method that prints a matrix in a readable way so that every column is correctly aligned and all elements are rounded to a fixed number of digits.

Example

The greater-than symbol followed by a space (> ) represents the user input. Note that it's not part of the input.

Example 1: the user chooses to calculate the inverse of a matrix

1. Add matrices
2. Multiply matrix by a constant
3. Multiply matrices
4. Transpose matrix
5. Calculate a determinant
6. Inverse matrix
0. Exit
Your choice: > 6
Enter matrix size: > 3 3
Enter matrix:
> 2 -1 0
> 0 1 2
> 1 1 0
The result is:
 0.33   0  0.33
-0.33   0  0.66
 0.16 0.5 -0.33

1. Add matrices
2. Multiply matrix by a constant
3. Multiply matrices
4. Transpose matrix
5. Calculate a determinant
6. Inverse matrix
0. Exit
Your choice: >

Example 2: the user chooses to calculate the inverse of a matrix; the matrix doesn't have an inverse

1. Add matrices
2. Multiply matrix by a constant
3. Multiply matrices
4. Transpose matrix
5. Calculate a determinant
6. Inverse matrix
0. Exit
Your choice: > 6
Enter matrix size: > 2 2
Enter matrix:
> 2 1
> 4 2
This matrix doesn't have an inverse.

1. Add matrices
2. Multiply matrix by a constant
3. Multiply matrices
4. Transpose matrix
5. Calculate a determinant
6. Inverse matrix
0. Exit
Your choice: > 0