Matrix multiplication (%*%) in R Programming - Step-by-Step Execution

Concept Flow - Matrix multiplication (%*%)

Start with Matrix A and Matrix B

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Check dimensions: cols of A == rows of B?

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Error: Cannot multiply

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Multiply rows of A by columns of B

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Sum products for each element

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Build Result Matrix

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End

Matrix multiplication takes two matrices, checks if they can be multiplied, then multiplies rows of the first by columns of the second, summing products to form the result.

Execution Sample

R Programming

A <- matrix(c(1,2,3,4), nrow=2)
B <- matrix(c(5,6,7,8), nrow=2)
C <- A %*% B
print(C)

Multiply two 2x2 matrices A and B using %*% and print the result.

Execution Table

Step	Action	Matrix A	Matrix B	Calculation	Result Matrix C
1	Define Matrix A	[1 3; 2 4]	[ ]	N/A	[ ]
2	Define Matrix B	[1 3; 2 4]	[5 7; 6 8]	N/A	[ ]
3	Check dimensions	2x2	2x2	cols A=2 == rows B=2	Proceed
4	Calculate C[1,1]	Row1 A: (1,3)	Col1 B: (5,6)	1*5 + 3*6 = 5 + 18 = 23	C[1,1] = 23
5	Calculate C[1,2]	Row1 A: (1,3)	Col2 B: (7,8)	1*7 + 3*8 = 7 + 24 = 31	C[1,2] = 31
6	Calculate C[2,1]	Row2 A: (2,4)	Col1 B: (5,6)	2*5 + 4*6 = 10 + 24 = 34	C[2,1] = 34
7	Calculate C[2,2]	Row2 A: (2,4)	Col2 B: (7,8)	2*7 + 4*8 = 14 + 32 = 46	C[2,2] = 46
8	Final Result	[1 3; 2 4]	[5 7; 6 8]	All elements calculated	[23 31; 34 46]

💡 All elements of result matrix C calculated; multiplication complete.

Variable Tracker

Variable	Start	After Step 4	After Step 5	After Step 6	After Step 7	Final
A	[1 3; 2 4]	[1 3; 2 4]	[1 3; 2 4]	[1 3; 2 4]	[1 3; 2 4]	[1 3; 2 4]
B	[ ]	[5 7; 6 8]	[5 7; 6 8]	[5 7; 6 8]	[5 7; 6 8]	[5 7; 6 8]
C	[ ]	[23 NA; NA NA]	[23 31; NA NA]	[23 31; 34 NA]	[23 31; 34 46]	[23 31; 34 46]

Key Moments - 3 Insights

Why must the number of columns in Matrix A equal the number of rows in Matrix B?

Because each element in the result is a sum of products between a row of A and a column of B, their lengths must match (see Step 3 in execution_table).

How is each element of the result matrix calculated?

Each element is the sum of multiplying corresponding elements from a row of A and a column of B (see Steps 4-7 in execution_table).

Why does the result matrix have dimensions equal to rows of A and columns of B?

Because each row of A combines with each column of B to produce one element, so result size is rows of A by columns of B (see Step 8).

Visual Quiz - 3 Questions

Look at the execution_table at Step 4. What is the value of C[1,1]?

A 34

B 23

C 31

D 46

💡 Hint

Check the 'Result Matrix C' column at Step 4 in execution_table.

At which step does the condition for matrix multiplication dimensions get checked?

A Step 3

B Step 2

C Step 5

D Step 7

💡 Hint

Look for the step mentioning 'Check dimensions' in execution_table.

If Matrix B had 3 rows instead of 2, what would happen at Step 3?

A Result matrix would be larger

B Multiplication proceeds normally

C Error because cols of A != rows of B

D No change in calculation

💡 Hint

Refer to the dimension check in Step 3 of execution_table.

Concept Snapshot

Matrix multiplication in R uses %*% operator.
Matrices A (m x n) and B (n x p) multiply if n matches.
Result is matrix C (m x p).
Each element C[i,j] = sum of A's row i * B's column j.
Use %*% for matrix product, not * which is element-wise.

Full Transcript

Matrix multiplication in R uses the %*% operator to multiply two matrices. First, the number of columns in the first matrix must equal the number of rows in the second matrix. Then, each element of the result matrix is calculated by multiplying elements from a row of the first matrix by elements from a column of the second matrix and summing these products. The result matrix has dimensions equal to the number of rows of the first matrix and the number of columns of the second matrix. This process is shown step-by-step in the execution table, where each element of the result matrix is computed. If the dimensions do not match, multiplication cannot proceed.