R Matrix Operations

Matrix operations go beyond simple indexing. R supports mathematical matrix operations used in linear algebra, statistics, and machine learning — all with clean, concise syntax.

Element-wise vs Matrix Multiplication

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

# Element-wise (each position multiplied independently)
A * B
#      [,1] [,2]
# [1,]   5   21
# [2,]  12   32

# True matrix multiplication (dot product)
A %*% B
#      [,1] [,2]
# [1,]  23   31
# [2,]  34   46

Matrix multiplication diagram (A %*% B):
A:  [1 3]   B:  [5 7]
    [2 4]       [6 8]

Result[1,1] = 1×5 + 3×6 = 23
Result[1,2] = 1×7 + 3×8 = 31
Result[2,1] = 2×5 + 4×6 = 34
Result[2,2] = 2×7 + 4×8 = 46

Transpose

M <- matrix(c(1,2,3,4,5,6), nrow=2)
#      [,1] [,2] [,3]
# [1,]   1    3    5
# [2,]   2    4    6

t(M)
#      [,1] [,2]
# [1,]   1    2
# [2,]   3    4
# [3,]   5    6

Determinant and Inverse

M <- matrix(c(4, 3, 3, 2), nrow=2)

det(M)       # -1  (determinant)
solve(M)     # inverse matrix

# Verify: M %*% solve(M) should give identity matrix
round(M %*% solve(M))
#      [,1] [,2]
# [1,]    1    0
# [2,]    0    1

Solving a System of Linear Equations

# 2x + y = 5
# x + 3y = 10

A <- matrix(c(2, 1, 1, 3), nrow=2)
b <- c(5, 10)

solve(A, b)   # x=1, y=3

Adding Rows and Columns

m1 <- matrix(1:4, nrow=2)
m2 <- matrix(5:8, nrow=2)

rbind(m1, m2)   # stack m2 below m1 (add rows)
cbind(m1, m2)   # place m2 right of m1 (add columns)

# Add a row of totals
col_totals <- colSums(m1)
rbind(m1, col_totals)

Apply Functions Across Rows or Columns

scores <- matrix(c(85,92,78,90,76,88,95,70,82), nrow=3)
rownames(scores) <- c("Student1","Student2","Student3")
colnames(scores) <- c("Math","Science","English")

# Row means (each student's average)
apply(scores, 1, mean)
# Student1  Student2  Student3
#   85.00     84.67     85.00

# Column means (each subject's average)
apply(scores, 2, mean)
# Math  Science  English
# 85.33   83.33    85.00

Eigenvalues and Eigenvectors

M <- matrix(c(4,1,2,3), nrow=2)
eig <- eigen(M)

eig$values    # eigenvalues:  5  2
eig$vectors   # eigenvectors (columns)

Correlation Matrix

# Create from a data matrix
data <- matrix(c(85,72,90,88,91,78,95,82,70,76), ncol=2)
colnames(data) <- c("Math","Science")

cor(data)
#           Math    Science
# Math     1.0000   0.9234
# Science  0.9234   1.0000

Practical: Portfolio Return Calculation

# 3 stocks, 4 months of returns (%)
returns <- matrix(c(2,3,1, -1,2,4, 3,1,2, 4,2,3),
                  nrow=4, byrow=TRUE)
colnames(returns) <- c("StockA","StockB","StockC")
rownames(returns) <- c("Jan","Feb","Mar","Apr")

# Average monthly return per stock
apply(returns, 2, mean)
# StockA StockB StockC
#    2.0    2.0    2.5

# Total return each month
rowSums(returns)
# Jan Feb Mar Apr
#   6   5   6   9

Matrix operations are the mathematical engine behind linear regression, principal component analysis, and neural networks. Every statistical model you build in R ultimately reduces to matrix computation under the hood.