Python program to multiply two matrices has been shown here. Matrix $[A]_{m \times n}$ can be multiplied with matrix $[B]_{p \times q}$, iff $n = p$ i.e. the number of columns of the first matrix should to be equal to the number of rows in the second matrix. The order of the resultant matrix $C$ would be $m \times q$.

Algorithm, pseudocode and time complexity of the program have also been shown below.

## 1. Algorithm for matrix multiplication

1. Take two matrice $A_{m\times n}$ and $B_{p\times q}$ as input.

2. Declare another matrix $C_{m\times q}$ and initailize it with 0.

3. Check if $n = p$

4. If step [3] is false then display "Matrices can not be multiplied!" and go to step [8]

5. If step [3] is true, then

6. Repeat for each $i \in [0, m - 1]$

6.1. Repeat for each $j \in [0, q - 1]$

6.1.1. Repeat for each $k \in [0, p - 1]$

6.1.1.1. Perform $C[i][j] = C[i][j] + A[i][k] * B[k][j]$

7. Display $C_{m\times q}$ as the resultant matrix.

8. Exit program.

## 2. Pseudocode for matrix multiplication

Input: Two matrices $A_{m\times n}$ and $B_{p\times q}$

Output: $A * B$

1. Procedure matrixMultiplication($A_{m\times n}$, $B_{p\times q}$):

2. Initialize $C_{m\times q} = 0$

3. If $n == p$:

4. Repeat for each $i \in [0, m-1]$

5. Repeat for each $i \in [0, q-1]$

6. Repeat for each $k \in [0, p-1]$

7. $C_{ij}\leftarrow C_{ij} + A_{ik} * B_{kj}$

8. Return $C_{m\times q}$

9. Else:

10. Return Matrices can not be multiplied!

11. End Procedure

## 3. Time Complexity for matrix multiplication

Time Complexity: O($n^3$)

Where $n$ is the row and column size of two matrices.

## 4. Python Program for matrix multiplication

Code has been copied
# ******************************************
#           alphabetacoder.com
# Python Program for Matrix Multiplication
# ******************************************

# declare a function which takes
# two matrices as inputs and returns
# the resultant matrix after multiplication
def matrixMultiplication(A, B):
# get the size of resultant matrix
x = len(A)
y = len(B[0])
# declare and initialize resultant matrix
C = [[0 for j in range(y)] for i in range(x)]
# do matrix multiplication
for i in range(0, x):
for j in range(0, y):
for k in range(0, len(B)):  # Use len(B) to get the number of rows in B
C[i][j] += A[i][k] * B[k][j]
return C

def main():
# take input of the order of first matrix
m = int(input("Enter the number of rows of the first matrix: "))
n = int(input("Enter the number of columns of the first matrix: "))

# take input of the order of second matrix
p = int(input("Enter the number of rows of the second matrix: "))
q = int(input("Enter the number of columns of the second matrix: "))

# check if number of columns in the first matrix
# is the same as the number of rows in the second matrix.
# If not, then matrices cannot be multiplied
if n != p:
print("Matrices cannot be multiplied!")
else:
# declare and initialize the matrices
A = [[0 for j in range(n)] for i in range(m)]
B = [[0 for j in range(q)] for i in range(p)]

# take input of the first matrix
print("Enter the first matrix of order", m, "x", n, ":")
for i in range(m):
for j in range(n):
A[i][j] = int(
input(f"Enter element at row {i + 1} and column {j + 1}: ")
)

# take input of the second matrix
print("Enter the second matrix of order", p, "x", q, ":")
for i in range(p):
for j in range(q):
B[i][j] = int(
input(f"Enter element at row {i + 1} and column {j + 1}: ")
)

# call the function to get the resultant matrix
# after matrix A is multiplied by matrix B
C = matrixMultiplication(A, B)

# display matrix C
print("The resultant matrix after multiplication: ")
# display the result
for i in range(m):
for j in range(q):
print(C[i][j], end=" ")
# new line
print("")

# drive code
main()


Output

Enter the number of rows of the first matrix: 3

Enter the number of columns of the first matrix: 3

Enter the number of rows of the second matrix: 3

Enter the number of columns of the second matrix: 3

Enter the first matrix of order 3 x 3 :

Enter element at row 1 and column 1: 5

Enter element at row 1 and column 2: 6

Enter element at row 1 and column 3: 1

Enter element at row 2 and column 1: 0

Enter element at row 2 and column 2: 2

Enter element at row 2 and column 3: 0

Enter element at row 3 and column 1: 3

Enter element at row 3 and column 2: 4

Enter element at row 3 and column 3: 6

Enter the second matrix of order 3 x 3 :

Enter element at row 1 and column 1: 0

Enter element at row 1 and column 2: 1

Enter element at row 1 and column 3: 2

Enter element at row 2 and column 1: 5

Enter element at row 2 and column 2: 3

Enter element at row 2 and column 3: 4

Enter element at row 3 and column 1: 5

Enter element at row 3 and column 2: 0

Enter element at row 3 and column 3: 6

The resultant matrix after multiplication:

35 23 40

10 6 8

50 15 58