Matrix Algebra (%*%, solve, t)

Matrix algebra in R is the set of tools — matrix multiplication with %*%, transpose with t(), and solving systems with solve() — that let you do real linear algebra on two-dimensional grids of numbers.

Learn Matrix Algebra (%*%, solve, t) in our free R course — a beginner-friendly interactive lesson with worked examples, a practice exercise and a quick…

Part of the free R course at LearnCodingFast — hands-on lessons with examples you run in your browser, plus practice exercises and a quick quiz.

By the end of this lesson you'll build matrices, multiply them the right way, transpose and invert them, and solve a system of linear equations in a single call.

What You'll Learn in This Lesson

1️⃣ Building Matrices

Use matrix() with a vector of values plus nrow and/or ncol . By default R fills down the columns ; add byrow = TRUE to fill across rows the way you'd read them. The dim() , nrow() , and ncol() helpers report the shape.

2️⃣ Multiplying and Transposing

The operator %*% performs true matrix multiplication, where each output cell is a row-times-column dot product. A plain * multiplies matching cells instead — useful, but a different operation. The transpose t() flips rows and columns.

Whenever you see a dimension-mismatch error from %*% , check that the inner dimensions agree: an m×n times an n×p gives an m×p result.

3️⃣ Inverses and Solving Systems

The single function solve() does double duty. Called with one matrix it returns the inverse; called as solve(A, b) it solves the linear system A x = b for the unknown vector x . Use det() to check invertibility and diag() to build an identity matrix.

Your turn. Fill in the # TODO blank, run it, and compare with the expected output.

Mini-Challenge: Crack the Prices

Two equations, two unknowns — exactly what solve(A, b) is built for. Remember to use byrow = TRUE so your coefficient matrix matches the way you wrote the equations.

Common Errors (and the fix)

• Using * when you meant %*% — ❌ A * B multiplies cell-by-cell. ✅ Use A %*% B for the matrix product.
• "non-conformable arguments" — ❌ inner dimensions don't match. ✅ For A %*% B , ncol(A) must equal nrow(B) .
• Inverting to solve a system — ❌ solve(A) %*% b is slower and less stable. ✅ Use solve(A, b) .
• Forgetting byrow — ❌ entries land in the wrong cells. ✅ Add byrow = TRUE when you list values row by row.

Pro Tips

• 💡 Stack matrices fast: rbind() stacks rows, cbind() glues columns side by side.
• 💡 crossprod(A) computes t(A) %*% A faster than writing it out — common in regression.
• 💡 Check shape early: a quick dim() before %*% saves you from conformability errors.

📋 Quick Reference — Matrix Algebra

Task

R Syntax

Create matrix

matrix(1:6, nrow = 2)

Matrix product

A %*% B

Transpose

t(A)

Inverse

solve(A)

Solve A x = b

solve(A, b)

Determinant

det(A)

Identity matrix

diag(n)

Frequently Asked Questions

🎉 Lesson Complete!

• ✅ matrix() builds a grid — control fill order with byrow
• ✅ %*% is the matrix product; * is elementwise
• ✅ t() transposes rows and columns
• ✅ solve(A) inverts; solve(A, b) solves a system
• ✅ det() checks invertibility; diag() makes an identity
• ✅ Next lesson: Text Wrangling with stringr

Practice quiz

Continue this course

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