(Algebra: multiply two matrices) Write a function to...

(Algebra: multiply two matrices) Write a function to multiply two matrices. The header of the function is: def multiplyMatrix $(a, b)$ To multiply matrix a by matrix b, the number of columns in a must be the same as the number of rows in $b$, and the two matrices must have elements of the same or compatible types. Let $c$ be the result of the multiplication. Assume the column size of matrix a is n. Each element $c_{i j}$ is $a_{i 1} \times b_{1 j}+a_{i 2} \times b_{2 j}+\ldots .+a_{i n} \times b_{n j} .$ For example, for two $3 \times 3$ matrices a and b, $c$ is $$\left(\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right) \times\left(\begin{array}{lll} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{array}\right)=\left(\begin{array}{lll} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{array}\right)$$ where $c_{i j}=a_{i 1} \times b_{1 j}+a_{i 2} \times b_{2 j}+a_{i 3} \times b_{3 j}$ Write a test program that prompts the user to enter two $3 \times 3$ matrices and displays their product. Here is a sample run:

(Algebra: multiply two matrices) Write a function to multiply two matrices. The header of the function is:
def multiplyMatrix $(a, b)$
To multiply matrix a by matrix b, the number of columns in a must be the same as the number of rows in $b$, and the two matrices must have elements of the same or
compatible types. Let $c$ be the result of the multiplication. Assume the column size of matrix a is
n. Each element $c_{i j}$ is $a_{i 1} \times b_{1 j}+a_{i 2} \times b_{2 j}+\ldots .+a_{i n} \times b_{n j} .$ For
example, for two $3 \times 3$ matrices a and b, $c$ is
$$\left(\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right) \times\left(\begin{array}{lll}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} & b_{32} & b_{33}
\end{array}\right)=\left(\begin{array}{lll}
c_{11} & c_{12} & c_{13} \\
c_{21} & c_{22} & c_{23} \\
c_{31} & c_{32} & c_{33}
\end{array}\right)$$
where $c_{i j}=a_{i 1} \times b_{1 j}+a_{i 2} \times b_{2 j}+a_{i 3} \times b_{3 j}$
Write a test program that prompts the user to enter two $3 \times 3$ matrices and displays their product. Here is a sample run:

Key Concepts

Nested Loop Algorithm

Implementing matrix multiplication typically involves using nested loops. The outer loops iterate over the rows of the first matrix and the columns of the second matrix, while an inner loop calculates the dot product to produce each element of the result.

Matrix Multiplication

Matrix multiplication is the process of combining two matrices to produce a new matrix. Each element of the resulting matrix is computed as the sum of products of corresponding elements from a row of the first matrix and a column of the second matrix.

Dimension Compatibility

For matrix multiplication to be valid, the number of columns in the first matrix must equal the number of rows in the second matrix. This property ensures that the dot product of corresponding rows and columns can be computed.

Dot Product

The dot product is a fundamental operation in matrix multiplication. It involves multiplying corresponding elements of a row from the first matrix and a column from the second matrix, then summing these products to yield a single element in the resulting matrix.

Function Implementation and Testing

When writing a function to perform matrix multiplication, it is crucial to define a clear function interface and include error checking to ensure dimension compatibility. Testing the function with sample inputs helps validate the correctness of the multiplication logic and overall functionality.

Transcript

0:00 Hello.

00:02 For this problem, you're being asked to write a function that will multiply two matrices and then write a program that will test a user, sorry, test that function by prompting a user to enter two matrices and displaying the product of those two matrices.

00:25 So i'm going to be writing my code in jupiter notebooks and i'm just going to talk through what i have written here.

00:32 So i've started by labeling this programming exercise 11 .6.

00:37 And i defined two functions here.

00:41 The first is multiply matrix that takes input matrices a and b.

00:47 And before i get too much further, i do want to just state that there are multiple ways of writing this.

00:54 This is just what made the most sense.

00:57 To me, but you might prefer to write it in a different way.

01:02 But we're going to take input matrix a and b.

01:07 And in the problem, we're specifically dealing with three by three matrices.

01:17 So i've written this program to only accept three by three matrices.

01:22 It's going to throw an error if it's a matrix of a different size.

01:27 You could actually make it more generic so that it could work for any matrix, any size matrices, as long as the dimensions work so that you can actually multiply those two matrices.

01:44 But i'm just going to work with three by three.

01:46 So we need to initialize a matrix c that's going to hold the product of a and b.

01:52 And i just initialized that with zero.

01:54 So this is a three by three matrix because those were the dimensions given in the problem.

02:03 And then now i'm going to multiply.

02:05 So in order to do that, i want to identify a row and column.

02:10 So we're going to be at a specific location of c.

02:13 And then we want to set that element equal to what it should be equal to based off of the example given in the in the textbook, sorry.

02:30 So for matrix c at location row column, a row zero times b zero column, plus a row one times b one column, plus a row two times b two column is going to be the result that should be at that location.

02:53 And then this function is just going to return that matrix c.

02:58 I wrote a second function to have a user input a matrix.

03:08 I found that that was just going to be easier than having to write all of this out twice.

03:14 So i've defined a function called inter matrix of n.

03:19 It's going to start by printing inter matrix n.

03:23 So it'll say inter matrix 1, inter matrix 2, intermatrix 2, in.

03:30 Then we are again initializing an empty list here.

03:35 So grid is equal to zero...