Advanced Matrix Calculator

Matrix Operations Explained

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are widely used in scientific fields such as physics, computer graphics, probability theory, statistics, calculus, numerical analysis, and more.

Matrix Dimensions

The dimensions of a matrix A are typically denoted as m × n, where m is the number of rows and n is the number of columns. Each element is identified by two subscripts, such as a_i,j, where i represents the row and j represents the column.

Matrix Operations

Matrix Addition

• Only matrices of the same dimensions can be added.
• Each corresponding element is added together.

Example:

A = [1 2]
     [3 4]
B = [5 6]
     [7 8]

Result:

C = [6 8]
     [10 12]

Matrix Subtraction

• Only matrices of the same dimensions can be subtracted.
• Each corresponding element is subtracted.

Example:

A = [1 2]
     [3 4]
B = [5 6]
     [7 8]

Result:

C = [-4 -4]
     [-4 -4]

Matrix Multiplication

Scalar Multiplication:

Each element is multiplied by the scalar value.

Example:

A = [1 2]
     [3 4], c = 5

Result:

[5 10]
[15 20]

Matrix-Matrix Multiplication:

• Number of columns in the first matrix must equal the number of rows in the second matrix.
• The result matrix will have dimensions (rows of first matrix) × (columns of second matrix).

Example:

A = [1 2 1]
     [3 4 1]
B = [5 6 1 1]
     [7 8 1 1]
     [1 1 1 1]

Result:

C = [20 23 4 4]
     [44 51 8 8]

Power of a Matrix

Only square matrices can be raised to a power. The power operation involves multiplying the matrix by itself repeatedly.

Example:

A = [1 3]
     [2 1]

A² = [7 6]
     [4 7]

Transpose of a Matrix

Swaps the rows and columns of a matrix.

Example:

A = [1 3]
     [2 1]
A^T = [1 2]
          [3 1]

Determinant of a Matrix

2×2 Matrix:

Determinant = ad – bc

Example:

A = [2 4]
     [6 8]
|A| = 2×8 – 4×6 = -8

3×3 Matrix:

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

4×4 Matrix:

Expands to minors, alternating signs. Each minor is a 3×3 determinant. This continues until reduced to 2×2 matrices.

Inverse of a Matrix

A matrix A has an inverse A^-1 if A × A^-1 = Identity Matrix I.

Identity Matrix Examples:

    2×2:  [1 0]
          [0 1]
    3×3:  [1 0 0]
          [0 1 0]
          [0 0 1]    

Inverse of 2×2 Matrix:

A^-1 = (1 / (ad – bc)) × [d -b; -c a]

Example:

A = [2 4]
     [3 7]
A^-1 = [3.5 -2]
         [-1.5 1]

Inverse of 3×3 Matrix:

Uses determinants and minors. Complex and typically solved with computational tools or advanced formulas.
Related Calculator:
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External Resources:
Matrix Calculator on Calculator.net