Matrix Arithmetic and Operations
Perform element-wise and matrix multiplication, addition, and subtraction.
Matrix Arithmetic and Operations is a free R Academy lesson on CoddyKit — lesson 3 of 4. You can read the complete lesson below for free — then practise it hands-on in the browser with a built-in code editor and a 24/7 AI tutor. It is part of the R Academy learning path, one of 4 lessons in the course, and your progress syncs across the web and the CoddyKit app.
Element-wise vs Matrix Operations
R distinguishes between element-wise operations (using +, -, *, /) and true matrix operations (using %*%). Understanding this difference is critical for correct linear algebra.
A <- matrix(c(1,2,3,4), nrow = 2)
B <- matrix(c(5,6,7,8), nrow = 2)
cat('A:
'); print(A)
cat('B:
'); print(B)
# Element-wise multiply
cat('A * B (element-wise):
'); print(A * B)
# Matrix multiply
cat('A %*% B (matrix mult):
'); print(A %*% B)Element-wise Addition and Subtraction
+ and - between two matrices of the same dimensions add or subtract corresponding elements. These are purely element-wise operations.
# Monthly revenue vs target (in thousands)
revenue <- matrix(c(120,135,110,98, 145,160,132,115),
nrow = 2, byrow = TRUE)
target <- matrix(c(130,130,130,130, 150,150,150,150),
nrow = 2, byrow = TRUE)
deviation <- revenue - target
cat('Revenue vs Target deviation:
')
print(deviation)
cat('Above target?:
')
print(revenue >= target)Element-wise Division and Multiplication
Similarly, * and / between matrices of equal size operate element by element. This is useful for scaling, normalising, or combining matrices element-wise.
# Scale a matrix by another (element-wise)
prices <- matrix(c(10, 20, 15, 25), nrow = 2)
quantities <- matrix(c(5, 3, 8, 2), nrow = 2)
total_cost <- prices * quantities
cat('Total cost per item:
')
print(total_cost)
# Normalise to row maximum
max_prices <- matrix(rep(apply(prices, 1, max), 2), nrow = 2)
norm_prices <- prices / max_prices
cat('Normalised prices:
')
print(norm_prices)Scalar Operations on Matrices
You can apply a scalar (single number) operation to every element of a matrix using standard arithmetic operators. The scalar is recycled to match the matrix dimensions.
m <- matrix(c(10, 20, 30, 40, 50, 60), nrow = 2, ncol = 3)
cat('Original:
'); print(m)
cat('m + 5:
'); print(m + 5)
cat('m * 2:
'); print(m * 2)
cat('m / 10:
'); print(m / 10)
cat('m ^ 2:
'); print(m ^ 2)True Matrix Multiplication %*%
A %*% B performs linear algebra matrix multiplication. For this to work, the number of columns in A must equal the number of rows in B. The result has dimensions (rows of A) x (cols of B).
# 2x3 times 3x2 = 2x2
A <- matrix(c(1,2,3, 4,5,6), nrow = 2, byrow = TRUE)
B <- matrix(c(7,8, 9,10, 11,12), nrow = 3, byrow = TRUE)
cat('A (2x3):
'); print(A)
cat('B (3x2):
'); print(B)
cat('A %*% B (2x2):
'); print(A %*% B)rowSums() and colSums()
rowSums(m) computes the sum of each row; colSums(m) computes the sum of each column. Both return named vectors if the matrix has dimension names.
sales <- matrix(
c(120,135,110,98, 145,160,132,115, 88,95,102,78),
nrow = 3, byrow = TRUE,
dimnames = list(c('Q1','Q2','Q3'), c('Jan','Feb','Mar','Apr'))
)
cat('Sales matrix:
'); print(sales)
cat('Quarterly totals:', rowSums(sales), '
')
cat('Monthly totals: ', colSums(sales), '
')rowMeans() and colMeans()
rowMeans(m) and colMeans(m) return the average of each row and column respectively. They are faster than applying mean() with apply() for this common task.
exam_scores <- matrix(
c(88,92,75,81, 79,85,90,73, 95,78,82,88),
nrow = 3, byrow = TRUE,
dimnames = list(c('Alice','Bob','Carol'), c('Math','Sci','Eng','Hist'))
)
cat('Student averages:
')
print(rowMeans(exam_scores))
cat('Subject averages:
')
print(colMeans(exam_scores))Comparison Operations on Matrices
Comparison operators (==, >, <=, etc.) applied to matrices return a logical matrix of the same dimensions. You can then use this mask for subsetting or counting.
scores <- matrix(c(88,45,92,61,73,55,84,79), nrow = 2)
cat('Scores:
'); print(scores)
# Which elements pass?
passing_mask <- scores >= 60
cat('Passing mask:
'); print(passing_mask)
cat('Number passing:', sum(passing_mask), '
')
cat('Passing scores:', scores[passing_mask], '
')apply() for Custom Row/Col Operations
apply(m, MARGIN, FUN) applies a function to rows (MARGIN=1) or columns (MARGIN=2). Use it when you need a custom aggregation beyond rowSums/colMeans.
m <- matrix(c(3,1,4,1,5,9,2,6,5,3,5,8), nrow = 3)
cat('Matrix:
'); print(m)
# Max of each row
cat('Row max: ', apply(m, 1, max), '
')
# Range of each column
cat('Col range:
')
print(apply(m, 2, range))
# Custom: count values > 4 per row
cat('Values > 4 per row:', apply(m, 1, function(x) sum(x > 4)), '
')Broadcasting a Vector Across Rows
When you add or multiply a vector to a matrix, R recycles the vector — cycling through it to match the matrix length. The recycling goes column by column, which can produce unexpected results if you want row-wise operations.
m <- matrix(c(10,20,30, 40,50,60), nrow = 2)
cat('Matrix:
'); print(m)
# Add a 3-element vector: R recycles column-by-column
v <- c(1, 2, 3)
cat('m + c(1,2,3):
'); print(m + v)
# For row-wise scaling, use sweep()
row_weights <- c(0.5, 2.0) # scale row 1 by 0.5, row 2 by 2
cat('Row-wise scaled:
'); print(sweep(m, 1, row_weights, '*'))Arithmetic Operations Summary
Key matrix operation functions in R:
+,-,*,/— element-wise arithmetic%*%— matrix multiplication (linear algebra)rowSums(),colSums()— row/column totalsrowMeans(),colMeans()— row/column averagesapply(m, 1, fn)/apply(m, 2, fn)— custom row/col aggregationsweep()— row/column-wise operations with a vector
m <- matrix(c(4,9,16,25), nrow = 2)
cat('Original:
'); print(m)
cat('sqrt (element-wise):
'); print(sqrt(m))
cat('rowSums:', rowSums(m), '
')
cat('colMeans:', colMeans(m), '
')Quick Check
What does A * B compute when A and B are 2x2 matrices in R?
Recap: Matrix Arithmetic
Great work! Key takeaways from this lesson:
+,-,*,/on matrices are element-wise%*%is true matrix multiplication (rows × columns)rowSums(),colSums(),rowMeans(),colMeans()aggregate over rows/columns- Comparison operators return a logical matrix of the same shape
apply(m, 1, fn)applies a function row-wise;apply(m, 2, fn)column-wise- Use
sweep()for vectorised row/column scaling
# Final: compute grade curve (add 5 to all scores, cap at 100)
raw <- matrix(c(88,55,92,73,60,45,78,95), nrow = 2)
curved <- pmin(raw + 5, 100) # pmin applies min element-wise
cat('Raw:
'); print(raw)
cat('Curved:
'); print(curved)Frequently asked questions
Is the “Matrix Arithmetic and Operations” lesson free?
Yes — the full text of “Matrix Arithmetic and Operations” is free to read here on the web, and the R Academy course includes 4 lessons in total. To practise it interactively (a built-in code editor and a 24/7 AI tutor) and unlock the rest of the R Academy course, upgrade to CoddyKit PRO.
What will I learn in “Matrix Arithmetic and Operations”?
Perform element-wise and matrix multiplication, addition, and subtraction. You practise R Academy with hands-on code you run directly in the browser, and a 24/7 AI tutor answers your questions as you work through the lesson.
Do I need any experience to start R Academy?
No prior experience is required. R Academy on CoddyKit is structured for beginners through advanced learners; this is — lesson 3 of 4, so you can start here or from the beginning and move at your own pace.
How long does the “Matrix Arithmetic and Operations” lesson take?
Most CoddyKit lessons take about 5–10 minutes. Each one is bite-sized and interactive, so you make steady progress and pick up exactly where you left off across the web and the app.
Can I write and run code in this R Academy lesson?
Yes. Every R Academy lesson includes a built-in code editor, so you write and run real code right in your browser and get instant AI feedback — no local setup required.
All lessons in this course
- Creating Matrices with matrix()
- Matrix Indexing and Subsetting
- Matrix Arithmetic and Operations
- Transposing and Reshaping Matrices