14 Matrices in R

What This Chapter Covers

This chapter is a complete tour of the matrix, R’s two-dimensional, single-type container. You will learn three ways to build one (matrix(), rbind(), cbind()), how rows and columns are stored and named, the four styles of indexing ([i, j], single-bracket, negative, logical), and how to update cells or whole rows and columns. You will see how arithmetic and the summary functions rowSums(), colSums(), rowMeans(), and colMeans() treat matrices, and the handful of linear-algebra operators that matter for applied work: transpose t(), matrix multiplication %*%, the determinant det(), and the matrix inverse solve(). You will also meet apply(), R’s apply-family entry for 2-D data. By the end of this chapter you will be able to build a matrix for any rectangular, single-type problem and manipulate it with confidence.

flowchart LR
    B["Build <br> matrix() / rbind() / cbind()"] --> M["A Matrix"]
    M --> I["Index <br> [i, j], negative, logical"]
    M --> AR["Arithmetic <br> element-wise + scalar recycling"]
    M --> LA["Linear Algebra <br> t() %*% det() solve()"]
    M --> AP["Row/Column summaries <br> rowSums() apply()"]
    classDef default fill:#004466,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;

14.1 What a Matrix Is

Core Concept: A Vector with Two Dimensions

A matrix is a vector whose elements are laid out in a rectangle of rows and columns. All cells share one atomic type, which is what distinguishes a matrix from a data frame. Internally, R stores a matrix as a single vector with a dim attribute (c(nrow, ncol)), filled by column by default.

Property	Matrix	Data Frame
Number of dimensions	2	2
All columns one type?	Yes.	No.
Typical use	Numeric grids, images, distances, correlations.	Survey-style tabular data.
Core functions	`matrix()`, `%*%`, `t()`, `solve()`.	`data.frame()`, `dplyr` verbs.

14.2 Building a Matrix

matrix(): the Direct Constructor

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rbind() and cbind(): Build from Vectors

rbind() stacks vectors as rows; cbind() stacks them as columns. Both functions coerce to a matrix if all inputs share a type.

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Naming Rows and Columns

Like vector names, row and column names are accessed and set with rownames() and colnames() (or with the single-object dimnames()).

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14.3 Indexing a Matrix

[row, col] is the Core Pattern

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Dropping and Keeping the Matrix Class

When you select a single row or column, R collapses the result to a plain vector by default. Pass drop = FALSE to keep a matrix.

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Negative and Logical Indexing

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Common Mistake: Forgetting the Comma

m[1] selects the first element treating the matrix as a flat vector; m[1, ] selects the whole first row. The comma is not optional.

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14.4 Modifying a Matrix

Replacing Cells, Rows, and Columns

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Adding a Row or a Column

Use rbind() to append a new row and cbind() to append a new column.

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14.5 Arithmetic on Matrices

Element-Wise by Default

Arithmetic operators work cell by cell, exactly as they do on vectors. Scalars are recycled. Two matrices of the same shape combine element by element.

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* is Element-Wise; %*% is Matrix Multiplication

A common slip from linear-algebra textbooks is to expect A * B to perform matrix multiplication. In R, * is element-wise. Use %*% for the standard row-by-column product.

14.6 Summarising Rows and Columns

Fast Built-Ins

Function	Returns
`rowSums(m)`, `colSums(m)`	One sum per row or per column.
`rowMeans(m)`, `colMeans(m)`	One mean per row or per column.
`apply(m, 1, FUN)`	Apply `FUN` to each row.
`apply(m, 2, FUN)`	Apply `FUN` to each column.

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Expert Insight: Prefer the Built-Ins Where They Exist

rowSums() and colSums() are written in C and can be an order of magnitude faster than apply(m, 1, sum). Use them when they exist; reach for apply() for custom functions like sd or an anonymous summary.

14.7 Linear-Algebra Essentials

The Four You Will Use Most

Operator / Function	Does
`t(A)`	Transpose.
`A %*% B`	Matrix multiplication.
`det(A)`	Determinant (square matrices).
`solve(A)`	Matrix inverse (for non-singular square `A`). `solve(A, b)` solves `A x = b`.

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Common Mistake: solve(A) on a Singular Matrix

If det(A) == 0, the matrix is singular and has no inverse. solve() will error. Always check det() (or use solve(A, b) instead of computing the full inverse) for numerical stability.

14.8 Reshaping and Combining

dim()<- for Quick Reshapes

Assigning to dim() reshapes an existing vector or matrix. The total number of cells must stay the same.

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Turning a Matrix Back into a Vector

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14.9 A Worked Example: A Monthly Sales Matrix

A Regional-Quarterly Report

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Every technique from the chapter is on display: construction with matrix(), named dimnames, index-based extraction, element-wise arithmetic for growth, rowSums() / colSums() for summaries, and cbind() / rbind() to append totals.

Summary

Concept	Description
Building Matrices
Matrix Defined	A matrix is a vector with two dimensions and one type
matrix() Constructor	Direct constructor specifying data, nrow, ncol, byrow
rbind() and cbind()	Build matrices from existing vectors row-wise or column-wise
Naming Rows and Columns	Use dimnames or rownames()/colnames() for labelled access
Indexing
[row, col] Indexing	The core access pattern; comma-separator is required
Negative Indexing	Negative indices drop the specified rows or columns
Logical Indexing	Boolean masks select elements that meet a condition
drop = FALSE	Prevents R from dropping a single-row or single-column matrix to a vector