Matrix Calculator
Matrix Operations with Step-by-Step Solutions
Matrix Size:
Operation:
Matrix A
➕
Matrix B
Examples:
Matrix Addition
⎡1 2⎤ + ⎡5 6⎤ = ⎡6 8⎤
⎣3 4⎦ ⎣7 8⎦ ⎣10 12⎦
Matrix Multiplication
⎡1 2⎤ × ⎡5 6⎤ = ⎡19 22⎤
⎣3 4⎦ ⎣7 8⎦ ⎣43 50⎦
Determinant (2×2)
|1 2| = 1×4 - 2×3 = -2
|3 4|
Transpose
⎡1 2⎤ᵀ = ⎡1 3⎤
⎣3 4⎦ ⎣2 4⎦
Frequently Asked Questions (FAQs)
A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used to represent linear transformations and solve systems of equations.
Matrices can only be added if they have the same dimensions. Add corresponding elements: (A + B)[i][j] = A[i][j] + B[i][j].
For multiplication A × B, the number of columns in A must equal number of rows in B. Result matrix has rows of A and columns of B.
The determinant is a scalar value that can be computed from a square matrix. It provides information about matrix properties like invertibility.
Transpose flips a matrix over its diagonal, switching row and column indices. If A has size m×n, Aᵀ has size n×m.
Matrix Operations Guide
Matrix Addition
- Matrices must have same dimensions
- Add corresponding elements
- Result matrix has same size
Matrix Multiplication
- Columns of A = Rows of B
- Dot product of rows and columns
- Result size: (rows of A) × (columns of B)
Determinant
- Only for square matrices
- 2×2: ad - bc
- 3×3: a(ei - fh) - b(di - fg) + c(dh - eg)
Transpose
- Swap rows and columns
- Aᵀ[i][j] = A[j][i]
- Size changes from m×n to n×m
Note: Matrix multiplication is not commutative. A × B ≠ B × A in most cases.
