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# Python program to multiply two matrices

Given two matrix the task is that we will have to create a program to multiply two matrices in python.

Examples:

```Input : X = [[1, 7, 3],
[3, 5, 6],
[6, 8, 9]]
Y = [[1, 1, 1, 2],
[6, 7, 3, 0],
[4, 5, 9, 1]]

Output : [55, 65, 49, 5]
[57, 68, 72, 12]
[90, 107, 111, 21]```

Using Simple Nested Loops: In this program we have to use nested for loops to iterate through each row and each column.

Implementation:

## Python3

 `# Program to multiply two matrices using nested loops`   `# take a 3x3 matrix` `A ``=` `[[``12``, ``7``, ``3``],` `    ``[``4``, ``5``, ``6``],` `    ``[``7``, ``8``, ``9``]]`   `# take a 3x4 matrix    ` `B ``=` `[[``5``, ``8``, ``1``, ``2``],` `    ``[``6``, ``7``, ``3``, ``0``],` `    ``[``4``, ``5``, ``9``, ``1``]]` `    `  `result ``=` `[[``0``, ``0``, ``0``, ``0``],` `        ``[``0``, ``0``, ``0``, ``0``],` `        ``[``0``, ``0``, ``0``, ``0``]]`   `# iterating by row of A` `for` `i ``in` `range``(``len``(A)):`   `    ``# iterating by column by B ` `    ``for` `j ``in` `range``(``len``(B[``0``])):`   `        ``# iterating by rows of B` `        ``for` `k ``in` `range``(``len``(B)):` `            ``result[i][j] ``+``=` `A[i][k] ``*` `B[k][j]`   `for` `r ``in` `result:` `    ``print``(r)`

Output:

```[114, 160, 60, 27]
[74, 97, 73, 14]
[119, 157, 112, 23]```

Time Complexity: O(M*M*N), as we are using nested loop traversing, M*M*N.
Auxiliary Space: O(M*N), as we are using a result matrix which is extra space.

Method 2: Matrix Multiplication Using Nested List. We use zip in Python.

Implementation:

## Python3

 `# Program to multiply two matrices using list comprehension`   `# take a 3x3 matrix` `A ``=` `[[``12``, ``7``, ``3``],` `    ``[``4``, ``5``, ``6``],` `    ``[``7``, ``8``, ``9``]]`   `# take a 3x4 matrix` `B ``=` `[[``5``, ``8``, ``1``, ``2``],` `    ``[``6``, ``7``, ``3``, ``0``],` `    ``[``4``, ``5``, ``9``, ``1``]]`   `# result will be 3x4` `result ``=` `[[``sum``(a ``*` `b ``for` `a, b ``in` `zip``(A_row, B_col)) ` `                        ``for` `B_col ``in` `zip``(``*``B)]` `                                ``for` `A_row ``in` `A]`   `for` `r ``in` `result:` `    ``print``(r)`

Output:

```[114, 160, 60, 27]
[74, 97, 73, 14]
[119, 157, 112, 23]```

Time Complexity: O(M*M*N), as we are using nested loop traversing, M*M*N.
Auxiliary Space: O(M*N), as we are using a result matrix which is extra space.

Method 3: Matrix Multiplication (Vectorized implementation).

Implementation:

## Python3

 `# Program to multiply two matrices (vectorized implementation)`   `# Program to multiply two matrices (vectorized implementation)` `import` `numpy as np` `# take a 3x3 matrix` `A ``=` `[[``12``, ``7``, ``3``],` `    ``[``4``, ``5``, ``6``],` `    ``[``7``, ``8``, ``9``]]`   `# take a 3x4 matrix` `B ``=` `[[``5``, ``8``, ``1``, ``2``],` `    ``[``6``, ``7``, ``3``, ``0``],` `    ``[``4``, ``5``, ``9``, ``1``]]`   `# result will be 3x4`   `result``=` `[[``0``,``0``,``0``,``0``],` `        ``[``0``,``0``,``0``,``0``],` `        ``[``0``,``0``,``0``,``0``]]`   `result ``=` `np.dot(A,B)`   `for` `r ``in` `result:` `    ``print``(r)`

Output:

```[114, 160, 60, 27]
[74, 97, 73, 14]
[119, 157, 112, 23]```

Time Complexity: O(M*M*N), as we are using nested loop traversing, M*M*N.
Auxiliary Space: O(M*N), as we are using a result matrix which is extra space.

Method 4:Using recursive matrix multiplication:

Algorithm:

1. Check if the dimensions of the two input matrices are valid for multiplication. If the number of columns in the first matrix does not equal the number of rows in the second matrix, then they cannot be multiplied. In this case, raise an error or return a None object.
2. Check if the matrices are of size 1×1. If they are, multiply their elements and return the result.
3. If the matrices are not of size 1×1, divide each matrix into four submatrices of equal size.
4. Recursively multiply the submatrices using the same algorithm, until each submatrix is of size 1×1.
5. Compute the product of the resulting submatrices using the formula:
6. result = [A11B11 + A12B21, A11B12 + A12B22, A21B11 + A22B21, A21B12 + A22B22]
7. Return the resulting matrix.

## Python3

 `def` `matrix_multiply_recursive(A, B):` `    ``# check if matrices can be multiplied` `    ``if` `len``(A[``0``]) !``=` `len``(B):` `        ``raise` `ValueError(``"Invalid matrix dimensions"``)`   `    ``# initialize result matrix with zeros` `    ``result ``=` `[[``0` `for` `j ``in` `range``(``len``(B[``0``]))] ``for` `i ``in` `range``(``len``(A))]`   `    ``# recursive multiplication of matrices` `    ``def` `multiply(A, B, result, i, j, k):` `        ``if` `i >``=` `len``(A):` `            ``return` `        ``if` `j >``=` `len``(B[``0``]):` `            ``return` `multiply(A, B, result, i``+``1``, ``0``, ``0``)` `        ``if` `k >``=` `len``(B):` `            ``return` `multiply(A, B, result, i, j``+``1``, ``0``)` `        ``result[i][j] ``+``=` `A[i][k] ``*` `B[k][j]` `        ``multiply(A, B, result, i, j, k``+``1``)`   `    ``# perform matrix multiplication` `    ``multiply(A, B, result, ``0``, ``0``, ``0``)` `    ``return` `result`     `# example usage` `A ``=` `[[``12``, ``7``, ``3``], [``4``, ``5``, ``6``], [``7``, ``8``, ``9``]]` `B ``=` `[[``5``, ``8``, ``1``, ``2``], [``6``, ``7``, ``3``, ``0``], [``4``, ``5``, ``9``, ``1``]]`   `result ``=` `matrix_multiply_recursive(A, B)` `for` `row ``in` `result:` `    ``print``(row)`

Output

```[114, 160, 60, 27]
[74, 97, 73, 14]
[119, 157, 112, 23]```

Time complexity: O(n^3)
Auxiliary Space : O(n^2)

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