Mastering The Row-Column Rule: A Step-By-Step Guide To Multiplying Matrices

Row-column rule to computing product of matrices

If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of the corresponding entries from row i of A and column j of B.

The row-column rule, also known as the dot product, is used to compute the product of two matrices. In this rule, we multiply the elements in a row of the first matrix with the corresponding elements in a column of the second matrix and add up the product.

Here are the steps to compute the product of matrices using the row-column rule:

Step 1: Check if the dimensions of the matrices are compatible for multiplication. The number of columns in the first matrix must be equal to the number of rows in the second matrix. If the dimensions are not compatible, the product is undefined.

Step 2: Begin with the first row of the first matrix and the first column of the second matrix.

Step 3: Multiply the corresponding elements in the row and column. For Example, if the element in the first row and first column are a1 and b1 respectively, then the product is a1*b1.

Step 4: Continue to multiply and add up each corresponding element in the row and column until all elements in the row and column have been exhausted.

Step 5: This will give a single element of the resulting matrix. Repeat the process for each element in the resulting matrix, until all elements are filled.

Step 6: The resulting matrix will have dimensions equal to the number of rows in the first matrix and the number of columns in the second matrix.

Example:

Let’s say we have two matrices A and B shown below:

A = [2, 3, 4]
[1, 0, 2]

B = [1, 5]
[3, 2]
[2, 1]

To compute the product AB, we use the row-column rule as follows:

First element in the result matrix R:
R11 = 2*1 + 3*3 + 4*2 = 18

Second element in the result matrix R:
R12 = 2*5 + 3*2 + 4*1 = 21

Third element in the result matrix R:
R13 = 0*1 + 1*3 + 2*2 = 7

Fourth element in the result matrix R:
R21 = 1*1 + 0*3 + 2*2 = 5

Fifth element in the result matrix R:
R22 = 1*5 + 0*2 + 2*1 = 7

Sixth element in the result matrix R:
R23 = 1*1 + 0*2 + 2*0 = 1

Therefore, the resulting matrix R is:

R = [18 21]
[5 7]
[7 1]

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