How do you multiply two matrices?

Matrix multiplication (not element-wise): each element of the result matrix is the dot product of a row from the first matrix and a column from the second. For 2×2: C[i][j] = A[i][0]×B[0][j] + A[i][1]×B[1][j].

How do you find the determinant of a 2×2 matrix?

For a 2×2 matrix [[a,b],[c,d]], the determinant = ad − bc. For example, det([[3,2],[1,4]]) = 3×4 − 2×1 = 12 − 2 = 10.

What does it mean if a matrix has no inverse?

A matrix with determinant = 0 is called singular and has no inverse. This means the matrix equations it represents have either no solution or infinitely many solutions.

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Matrix Calculator

Add, subtract, multiply matrices. Calculate determinant, transpose, and inverse for 2×2 and 3×3 matrices with step-by-step working.

Matrix Size:

Operation

Matrix A

Matrix B

Result

Result Matrix

How it works

Result = A + B (element-wise addition)

Operation Reference

Add / Subtract: Element-wise, same size matrices

Multiply: C[i][j] = Σ A[i][k] × B[k][j]

Determinant 2×2: ad − bc

Determinant 3×3: Cofactor expansion along row 1

Transpose: Result[i][j] = A[j][i]

Inverse 2×2: (1/det) × [[d, −b], [−c, a]] — requires det ≠ 0

How to Use Matrix Calculator

1
Select the matrix size (2×2 or 3×3).
2
Select the operation you want to perform.
3
Enter values into Matrix A (and Matrix B for add/subtract/multiply).
4
The result appears instantly in the result grid.
5
Read the step-by-step calculation below the result.

About Matrix Calculator

The Matrix Calculator performs common matrix operations on 2×2 and 3×3 matrices: addition, subtraction, multiplication, determinant, transpose, and inverse. Each operation includes a step-by-step explanation of the calculation.

Supported Operations

Add/Subtract: Element-wise addition or subtraction of two matrices of the same size. Multiply: Standard matrix multiplication (dot product). Determinant: Scalar value det(A) — 2×2 uses ad−bc; 3×3 uses cofactor expansion along the first row. Transpose: Flip rows and columns. Inverse: For 2×2 matrices, (1/det) × [[d,−b],[−c,a]].

When is a Matrix Invertible?

A matrix is invertible (non-singular) only if its determinant is non-zero. If det = 0, the matrix is singular and has no inverse — the calculator will show an appropriate message.