Matrix Calculator
Select the rows and columns of the matrix and enter its elements to calculate all its properties, or access the step-by-step calculators for specific operations.
Quick Examples
How to Use the Calculator
This online matrix solver is a linear algebra tool designed to fully analyze any rectangular array as well as to direct you to specific arithmetic operations.
To get started, use the main menu to select the working mode you need:
1. Analyze a Matrix
This is the default mode and works like a matrix scanner that will give you all the properties of the array in a single report. To use it, follow these steps:
- Set the size: use the "rows" and "columns" selectors to define the dimension. Values from 1 to 8 are accepted in both selectors.
- Enter the values: you can use integers, decimals, negative numbers, or exact fractions. The algebraic engine processes the data while keeping exact fractions to avoid rounding errors.
Once the data is processed, the tool will generate a detailed report divided according to the matrix size:
- For any type of matrix (rectangular or square): you will get the original tabulated matrix, its dimension, the rank and nullity, the reduced row echelon form, and the transpose matrix.
- Exclusive to square matrices: if the number of rows matches the number of columns, the algorithm will additionally calculate the determinant (|A|), the trace (tr(A)), the adjoint matrix, and will verify its invertibility to provide the inverse matrix (if it exists).
In addition, the calculator will evaluate the array to classify it if it meets any special typology (symmetric, skew-symmetric, diagonal, triangular, identity, zero, idempotent, or involutory matrix).
When viewing the report, you will notice that most of the results (such as rank, inverse, or determinant) are interactive links. If you click on any of them, the system will automatically open a specialized calculator that will show you the step-by-step calculation of that property, keeping the same data you had already entered.
2. Step-by-Step Calculations
If you do not require a general analysis, but rather need to solve a specific problem from your homework, select the "Step-by-step calculations (operations and properties)" mode in the main menu. Doing so will display an organized panel with shortcuts to all the specific calculators, both for performing matrix operations and for calculating properties and transformations. These include:
Solved Examples
The following are examples of problems solved by the calculator.
Example 1
Analyze the following 3x3 matrix:
$$ \displaystyle A=\begin{bmatrix} 1 & 2 & 3 \\[0.5em] 2 & 4 & 5 \\[0.5em] 3 & 5 & 6 \end{bmatrix} $$
Complete analysis
Dimension: 3×3 (square)
Rank: 3
Nullity: 0
Reduced row echelon form:
Transpose matrix:
Determinant: \( \displaystyle |A| = -1 \)
Trace: \( \displaystyle \text{tr}(A) = 11 \)
Invertibility: The matrix is invertible (nonsingular).
Inverse matrix:
Adjoint matrix:
Classification:
- Symmetric matrix (the matrix is equal to its transpose).
Example 2
Determine the properties of the 2x2 matrix:
$$ \displaystyle A=\begin{bmatrix} \frac{1}{2} & -\frac{1}{4} \\[0.5em] 2 & 3 \end{bmatrix} $$
Complete analysis
Dimension: 2×2 (square)
Rank: 2
Nullity: 0
Reduced row echelon form:
Transpose matrix:
Determinant: \( \displaystyle |A| = 2 \)
Trace: \( \displaystyle \text{tr}(A) = \frac{7}{2} \)
Invertibility: The matrix is invertible (nonsingular).
Inverse matrix:
Adjoint matrix:
Example 3
Analyze the following 3x3 matrix:
$$ \displaystyle A=\begin{bmatrix} 1 & -1 & 2 \\[0.5em] 0 & 2 & 1 \\[0.5em] 3 & 0 & 4 \end{bmatrix} $$
Complete analysis
Dimension: 3×3 (square)
Rank: 3
Nullity: 0
Reduced row echelon form:
Transpose matrix:
Determinant: \( \displaystyle |A| = -7 \)
Trace: \( \displaystyle \text{tr}(A) = 7 \)
Invertibility: The matrix is invertible (nonsingular).
Inverse matrix:
Adjoint matrix:
Example 4
Determine the properties of the following 3x3 matrix:
$$ \displaystyle A=\begin{bmatrix} 1 & 2 & 3 \\[0.5em] 0 & 4 & 5 \\[0.5em] 0 & 0 & 6 \end{bmatrix} $$
Complete analysis
Dimension: 3×3 (square)
Rank: 3
Nullity: 0
Reduced row echelon form:
Transpose matrix:
Determinant: \( \displaystyle |A| = 24 \)
Trace: \( \displaystyle \text{tr}(A) = 11 \)
Invertibility: The matrix is invertible (nonsingular).
Inverse matrix:
Adjoint matrix:
Classification:
- Upper triangular matrix (the elements below the main diagonal are zero).
Example 5
Analyze the 3x4 matrix given below:
$$ \displaystyle A=\begin{bmatrix} 1 & 0 & -2 & 3 \\[0.5em] 2 & 1 & 4 & 0 \\[0.5em] 0 & -1 & 1 & 2 \end{bmatrix} $$
Complete analysis
Dimension: 3×4
Rank: 3
Nullity: 1
Reduced row echelon form:
Transpose matrix:
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