Matrices in MV Workstation
Type a matrix with square brackets, commas between entries and ; between rows: [1, 2; 3, 4]. Add, subtract, multiply, find determinants, inverses and transposes exactly, with letters in the entries too, and see the working step by step.
Type the maths the way you write it and MV Workstation shows the exact answer, the working and, where it helps, a picture you can drag.
Things to try
| What it is | Type this |
|---|---|
| A 2×2 matrix | [1, 2; 3, 4] |
| Entries with letters | [1, 2; 2, x + 1] |
| Multiply matrices | [1, 2; 3, 4] * [5, 6; 7, 8] |
| Determinant | det([1, 2; 3, 4]) |
| Inverse | inverse([2, 1; 1, 1]) |
| Transpose | [1, 2, 3; 4, 5, 6]^T |
| A row and a column | [1, 2, 3] * [4; 5; 6] |
| Name a matrix | A = [1, 2; 3, 4] |
| Row reduce | rref([1, 2, 3; 4, 5, 6; 7, 8, 10]) |
| Solve A x = b | linsolve([2, 1; 1, 3], [5; 10]) |
| Eigenvalues | eigenvalues([2, 1; 1, 2]) |
| Eigenvalues of a 3×3 | eigenvalues([2, 0, 0; 0, 3, 4; 0, 4, 9]) |
| A x = b | [2, 1; 1, 3] x = [5; 10] |
Tools
- Matrix steps: Show the working for a determinant, inverse, product, sum, power, transpose, trace, row reduction, rank, linear system or eigenvalues, one step at a time.
Pictures you can drag
- What a 2×2 matrix does: See where the unit square and the two basis arrows go. Drag the arrow tips and the matrix follows.
Functions
| Function | What it does | Example |
|---|---|---|
det(A) | The determinant of a square matrix. determinant(A) means the same. | det([1, 2; 3, 4]) |
inv(A) | The inverse of a square matrix: A * inv(A) is the identity. inverse(A) and A^(-1) mean the same. A matrix with determinant 0 has none. | inv([1, 2; 3, 4]) |
transpose(A) | Swap rows and columns. A^T means the same. | transpose([1, 2, 3; 4, 5, 6]) |
trace(A) | The sum of the entries on the main diagonal of a square matrix. tr(A) means the same. | trace([1, 2; 3, 4]) |
identity(n) | The n × n identity matrix: ones on the main diagonal, zeros elsewhere. | identity(3) |
rref(A) | The reduced row echelon form: row operations until every pivot is 1 with zeros above and below. Needs a matrix of plain numbers. rowreduce(A) means the same. | rref([1, 2, 3; 4, 5, 6; 7, 8, 10]) |
rank(A) | The rank: how many independent rows the matrix has (the number of pivots after row reduction). | rank([1, 2; 2, 4]) |
linsolve(A, b) | Solve A x = b for a column b, by row reduction. Says so when there is no solution or infinitely many. | linsolve([2, 1; 1, 3], [5; 10]) |
eigenvalues(A) | The eigenvalues of a 2×2 or 3×3 matrix, as a row, exact (square roots stay as square roots; a 3×3 whose cubic has no neat root shows decimals). eigvals(A) means the same. | eigenvalues([2, 1; 1, 2]) |
eigenvectors(A) | The eigenvectors of a 2×2 or 3×3 matrix, one per column, in the same order as eigenvalues(A). For a 3×3 they are exact when every eigenvalue is a whole number or fraction. eigvecs(A) means the same. | eigenvectors([2, 1; 1, 2]) |
Open MV Workstation, it is free
MV Workstation is a visual maths whiteboard that solves, shows the steps and draws as you type. It runs in your browser and works offline.