An online matrix calculator removes the arithmetic grind from linear algebra: enter your numbers once and read the determinant, transpose and inverse straight back. This guide shows exactly how our free matrix calculator takes input, what it returns, and how to read every result correctly — up to 3×3.
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Entering a matrix
Switch to MATRIX mode. Type each row on its own line, with the entries in a row separated by spaces. For example, this is the 2×2 matrix A = [[1, 2], [3, 4]]:
1 2
3 4As soon as Matrix A is valid, the calculator displays its determinant, transpose (Aᵀ) and inverse (A⁻¹) together. Enter an optional Matrix B to also get A + B and A × B.
Determinant
The determinant is a single number that tells you whether a matrix is invertible (non-zero) or singular (zero), and scales how the matrix stretches area or volume.
A = [[1, 2], [3, 4]] → det(A) = −2. By hand: (1·4) − (2·3) = 4 − 6 = −2. ✔
A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]] → det(A) = 1. Because it's non-zero, this matrix has an inverse.
Want to see the by-hand method behind that number? See how to find a 3×3 determinant by hand vs calculator.
Transpose
The transpose Aᵀ flips a matrix across its main diagonal: row 1 becomes column 1, and so on.
A = [[1, 2, 3], [4, 5, 6]] → Aᵀ = [[1, 4], [2, 5], [3, 6]]. A 2×3 becomes a 3×2.
Inverse
The inverse A⁻¹ undoes A: A × A⁻¹ equals the identity matrix. It exists only when det(A) ≠ 0.
A = [[1, 2], [3, 4]] → A⁻¹ = [[−2, 1], [1.5, −0.5]]. Check: multiply by A and you get the identity.
A = [[1, 2], [2, 4]] has det(A) = 0 (the rows are proportional), so it has no inverse and the inverse step can't be computed.
Adding & multiplying
Enter Matrix B to unlock A + B and A × B.
- Addition needs A and B to be the same size; it adds entry by entry.
- Multiplication needs the columns of A to equal the rows of B, and is not commutative.
A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]] → A × B = [[19, 22], [43, 50]]. The top-left entry is the dot of row 1 of A with column 1 of B: 1·5 + 2·7 = 19.
The full row-by-column method is in the matrix multiplication walkthrough. To solve linear systems with these tools, see solving simultaneous equations with matrices. For 3D vectors, switch to the vector dot & cross product guide.
Frequently asked questions
How do I enter a matrix?
In MATRIX mode, type each row on its own line with entries separated by spaces. The determinant, transpose and inverse appear automatically.
What size of matrix is supported?
Up to 3×3, matching the Casio fx-991ES PLUS.
Why does the inverse show an error?
The determinant is zero — a singular matrix has no inverse.
Is matrix multiplication commutative?
No — in general A × B ≠ B × A, and the inner dimensions must match.
Try the matrix calculator
Open it, switch to MATRIX mode, and paste the rows of any example above.
Open the online matrix calculator →