Matrix Calculator

About Matrix Calculator

Matrix Operations Reference

The six operations supported by the calculator each have different dimensional requirements and produce different shapes of output. Check this table before entering data so the calculator does not reject your matrices.

How Does Matrix Multiplication Work?

To multiply A (m×n) by B (n×p), each entry of the result is the dot product of a row from A with a column from B. The rule comes directly from the definition of a linear transformation composed with another linear transformation, as set out in Gilbert Strang's MIT 18.06 linear algebra notes.

C[i][j] = Σ A[i][k] × B[k][j]

Worked example. A = [1, 2; 3, 4], B = [5, 6; 7, 8]:

Result: C = [19, 22; 43, 50].

Non-commutativity. Matrix multiplication is not commutative. Using the same A and B above, B × A = [23, 34; 31, 46], which is different from A × B. This is a real algebraic difference, not a quirk of notation - multiplying a rotation by a scaling gives a different result than multiplying a scaling by a rotation, which is why 3D graphics pipelines are strict about the order in which transforms are applied.

How to Calculate a Determinant

The determinant of a square matrix is a single number that encodes whether the matrix has an inverse (det ≠ 0) and how much the matrix scales area or volume as a linear map. For a 2×2 matrix use the diagonal rule; for 3×3 and larger, use cofactor expansion along any row or column.

2×2 matrix:

det([a, b; c, d]) = ad - bc

Example: det([3, 7; 1, 5]) = 3×5 - 7×1 = 15 - 7 = 8.

3×3 matrix (cofactor expansion along the first row):

det(A) = a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₂₁a₃₃ - a₂₃a₃₁) + a₁₃(a₂₁a₃₂ - a₂₂a₃₁)

Worked example. A = [2, 1, 3; 0, -1, 2; 1, 4, -1]:

1. 2 × det([-1, 2; 4, -1]) = 2 × (1 - 8) = 2 × (-7) = -14
2. -1 × det([0, 2; 1, -1]) = -1 × (0 - 2) = -1 × (-2) = 2
3. 3 × det([0, -1; 1, 4]) = 3 × (0 + 1) = 3 × 1 = 3
4. det(A) = -14 + 2 + 3 = -9

The calculator shows this full cofactor breakdown for 2×2 and 3×3 matrices. For 4×4 and 5×5, it recursively expands along the first row until it reaches 2×2 blocks, which matches the Laplace expansion used in standard linear algebra textbooks.

What Does the Matrix Inverse Mean?

The inverse of a square matrix A, written A⁻¹, is the unique matrix that satisfies A × A⁻¹ = A⁻¹ × A = I, where I is the identity. An inverse exists exactly when det(A) ≠ 0. Geometrically, if A stretches and rotates vectors, then A⁻¹ is the transform that undoes it.

2×2 inverse formula:

A⁻¹ = (1/det(A)) × [d, -b; -c, a]

Worked example. A = [4, 7; 2, 6], det = 24 - 14 = 10.

• A⁻¹ = (1/10) × [6, -7; -2, 4] = [0.6, -0.7; -0.2, 0.4]
• Check: A × A⁻¹ = [4×0.6 + 7×(-0.2), 4×(-0.7) + 7×0.4; 2×0.6 + 6×(-0.2), 2×(-0.7) + 6×0.4] = [1, 0; 0, 1]. Correct.

For matrices larger than 2×2, the calculator uses Gauss-Jordan elimination with partial pivoting. It appends the identity to the right of A, row-reduces until the left half becomes the identity, and the right half is then A⁻¹. This is the method taught in Anton and Rorres' Elementary Linear Algebra and is numerically more stable than computing the classical adjugate for larger matrices.

When det(A) = 0, the matrix is singular. In that case the rows (or columns) are linearly dependent, the transform collapses at least one dimension to zero, and there is no way to undo it. The calculator displays an explicit singular-matrix message rather than returning garbage numbers.

Properties of Matrix Operations

Matrix arithmetic follows a small set of rules that are often tested in exams and used every day in applied work. The following table lists the identities you can rely on, plus the key one that catches people out: multiplication does not commute.

Where Are Matrices Used in the Real World?

Matrices show up any time you have to apply the same linear operation to many pieces of data at once. Modern graphics cards, machine learning libraries, and spreadsheet solvers are all matrix engines underneath.

Common Mistakes and Edge Cases

Matrix problems fail in predictable ways. These are the traps the calculator warns about and that students lose marks for in exams.

• Swapping row and column counts. A 2×3 matrix has 2 rows and 3 columns. Row count comes first. Getting this wrong turns every dimension check into guesswork.
• Trying to multiply incompatible shapes. 2×3 times 2×3 does not work. You need 2×3 times 3×k. The calculator shows the exact reason when it rejects a multiplication.
• Assuming AB = BA. Even for square matrices of the same size, AB and BA are usually different matrices. Rotate-then-scale is not the same as scale-then-rotate.
• Asking for the inverse of a singular matrix. If det(A) = 0, no inverse exists. Check the determinant first; a result of 0 (or very close to it, like 1e-12 from floating point) means stop.
• Floating-point determinants close to zero. Because the inverse uses Gauss-Jordan elimination, the calculator treats |det| < 1e-10 as singular to avoid dividing by near-zero pivots and producing huge, meaningless numbers.
• Forgetting (AB)⁻¹ = B⁻¹ A⁻¹. The order flips. This is easy to miss and is a common exam trap.

Tips for Using This Calculator Efficiently

A few small habits make matrix work much faster and reduce arithmetic errors.

• Set the size before typing values. Changing the row or column count after entering data preserves overlapping cells, but resizing larger pads with zeros and resizing smaller discards the extras. Set the dimensions first.
• Use Identity as a starting point. The Identity button gives you a matrix that already satisfies A × I = A, so it is a useful sanity check - multiply any matrix by the identity and the result should match the input.
• Check the determinant before asking for the inverse. If det(A) is zero or very small, the inverse is unstable or does not exist. The determinant step-by-step view also helps you spot a row or column of zeros instantly.
• Use integers where possible. Floating-point arithmetic in any matrix tool (including this one) will show small rounding artefacts like 0.9999999 instead of 1. Working with whole numbers where you can makes the output cleaner.
• Transpose is a shape check. If you are not sure which way round your data is, transposing it is cheap and shows you the alternative shape. This is especially helpful when lining up A and B for multiplication.

For solving systems of linear equations (which matrices represent compactly), the equation solver handles 2×2 systems using Cramer's rule - which is itself a ratio of determinants. To visualise mathematical functions alongside matrix work, the graphing calculator plots any expression, and the scientific calculator handles the arithmetic inside each cell if you want to compute entries by hand before entering them.

Every calculation runs in your browser. No matrix data is sent to any server.

Sources

• MIT - Gilbert Strang, Introduction to Linear Algebra
• MIT OpenCourseWare - 18.06 Linear Algebra
• Wolfram MathWorld - Matrix Inverse
• Wolfram MathWorld - Determinant
• Nobel Prize - Wassily Leontief (Input-Output Analysis)
• Nick Higham (University of Manchester) - What Is a Matrix Inverse?