Matrix & Determinant Calculator

Everything in the Class 11 and 12 matrices chapter in one place — operations, determinants expanded term by term, minors, cofactors, adjoint, inverse and Cramer's rule. Every answer stays an exact fraction, never a rounded decimal.

Class 11–12 Up to 4×4 Exact fractions Full working

For a 2 × 2 matrix [[a, b], [c, d]] the determinant is ad − bc and the inverse is (1 ÷ |A|) × [[d, −b], [−c, a]]. For [[1, 2], [3, 4]], |A| = 4 − 6 = −2, so the inverse is [[−2, 1], [1.5, −0.5]]. An inverse exists only when |A| is not 0.

Matrix A

Whole numbers, decimals (0.75) or fractions (3/4, −2/5).

×

Matrix B

Needed for A + B, A − B, A × B and B × A.

×
For k·A — fractions allowed.
For Aⁿ (A must be square).

Try an example

Loaded straight into A and B.

Choose an operation

The result and the working update as you type.

How it works

Determinants and inverses, line by line

Board exams give marks for the working, not just the answer. This calculator writes out every sign, minor and cofactor exactly the way you must in your answer sheet.

Set the order

Choose rows × columns for A (and B), up to 4 × 4.

Type the entries

Whole numbers, decimals or fractions — 0.6 and 3/5 mean the same thing.

Pick a tab

Operations, Determinant, Inverse, Cramer's rule or Properties.

Copy the steps

Copy the working as text, or print the page for your notebook.

Expanding a determinant

|A| = ai1Ai1 + ai2Ai2 + … + ainAin

Pick any one row (or column) and multiply each element by its cofactor. For a 3 × 3 along the first row:

|A| = a(ei − fh) − b(di − fg) + c(dh − eg)

The signs follow the chessboard pattern + − + / − + − / + − +.

Minor and cofactor

The minor Mij is the determinant left after deleting row i and column j. The cofactor adds the sign:

Aij = (−1)i+j × Mij

Example: in [[1,2,3],[4,5,6],[7,8,10]], M12 = |4 6; 7 10| = 40 − 42 = −2, so A12 = (−1)3 × (−2) = +2.

Adjoint and inverse

adj A is the transpose of the cofactor matrix, and

A−1 = adj A ÷ |A|

So A−1 exists only when |A| ≠ 0. A matrix with |A| = 0 is called singular and has no inverse. Useful check: A · adj A = |A| · I.

Cramer's rule

For a1x + b1y + c1z = d1 (and two more),

x = Dx/D, y = Dy/D, z = Dz/D

D is the determinant of the coefficients; Dx replaces the x-column with the constants. If D ≠ 0 there is exactly one solution. If D = 0 and some Dx, Dy, Dz is not 0 the system is inconsistent; if all are 0 it is either dependent (infinitely many solutions) or inconsistent — this tool checks the ranks and tells you which.

Order rules for ×

Am×n × Bn×p is defined only when the columns of A equal the rows of B, and the answer is m × p.

Each entry is a row-by-column sum: cij = Σ aikbkj.

Matrix multiplication is not commutative — A × B and B × A are usually different, and one may not even exist.

Special matrices

  • Symmetric: A′ = A (mirror across the diagonal)
  • Skew-symmetric: A′ = −A, so every diagonal entry is 0
  • Singular: |A| = 0, no inverse
  • Scalar: diagonal with all diagonal entries equal

A skew-symmetric matrix of odd order always has determinant 0 — try the example and see.

Exam tip: before expanding a 4 × 4, use row operations to make a row or column mostly zeros — the value of the determinant does not change when you add a multiple of one row to another, and every zero removes a whole 3 × 3 minor from your working.
Questions & answers

Matrix & Determinant Calculator — your questions

How do you find the determinant of a 3 by 3 matrix?

Expand along any row or column, multiplying each element by its cofactor. Along the first row of [[a, b, c], [d, e, f], [g, h, i]] this gives a(ei − fh) − b(di − fg) + c(dh − eg). The signs follow the chessboard pattern plus, minus, plus. Choosing a row with zeros makes the work much shorter.

What is the difference between a minor and a cofactor?

The minor Mij is the determinant left after deleting row i and column j. The cofactor adds a sign: Aij = (−1)^(i+j) × Mij. In [[1, 2, 3], [4, 5, 6], [7, 8, 10]], M12 is the determinant of [[4, 6], [7, 10]] = 40 − 42 = −2, so A12 = −1 × −2 = 2.

How do I find the inverse of a matrix?

The inverse is A⁻¹ = adj A ÷ |A|, where adj A is the transpose of the matrix of cofactors. So compute every cofactor, transpose that matrix, then divide each entry by the determinant. A useful check is A × adj A = |A| × I, which this tool shows alongside the answer.

What is a singular matrix?

A square matrix is singular when its determinant is 0. Such a matrix has no inverse, because the inverse formula divides by the determinant. In a system of equations, a singular coefficient matrix means there is either no solution or infinitely many, never exactly one.

How does Cramer's rule solve a system of equations?

Write D for the determinant of the coefficients, then replace the x column with the constants to get Dx, and similarly for Dy and Dz. The solution is x = Dx/D, y = Dy/D, z = Dz/D. If D is not 0 there is exactly one solution; if D = 0 the system is either inconsistent or dependent.

When can two matrices be multiplied?

A matrix of order m by n can be multiplied by one of order n by p only when the number of columns of the first equals the number of rows of the second, and the product has order m by p. Each entry is a row-by-column sum. Matrix multiplication is not commutative, so A × B and B × A usually differ.

What are symmetric and skew-symmetric matrices?

A matrix is symmetric when its transpose equals itself, so the entries mirror across the leading diagonal. It is skew-symmetric when its transpose equals its negative, which forces every diagonal entry to be 0. A skew-symmetric matrix of odd order always has determinant 0, and any square matrix can be written as the sum of a symmetric and a skew-symmetric matrix.