Linear Algebra Calculator

Perform matrix operations including determinants, inverses, multiplication, and more.

Linear Algebra is the branch of mathematics concerning linear equations, linear functions, and their representations in vector spaces and through matrices. It is fundamental to modern mathematics and essential in fields ranging from physics and engineering to computer science and data analysis.

This calculator allows you to perform common matrix operations on matrices of up to 5x5 size. You can calculate properties of a single matrix like the determinant or inverse, or perform arithmetic operations between two matrices.

Matrix Calculator

Matrix A

Size:x

Operation:

Result

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What is Linear Algebra Calculator?

Linear Algebra is a sub-field of mathematics that deals with vectors, matrices, and linear transforms. Unlike standard algebra which deals with scalars (single numbers), linear algebra solves systems of linear equations and studies the properties of multi-dimensional objects.

Formula & Calculation

Common Matrix Properties (for 2x2 Matrix A)

Determinant

For

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

\det(A) = ad - bc

Inverse

Only exists if $\det(A) \neq 0$ .

A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Example Calculation

Example: Matrix Multiplication

Let

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

and

B = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix}

To find $C = A \times B$ :

$c_{11} = (1)(2) + (2)(1) = 4$
$c_{12} = (1)(0) + (2)(2) = 4$
$c_{21} = (3)(2) + (4)(1) = 10$
$c_{22} = (3)(0) + (4)(2) = 8$

Result:

\begin{pmatrix} 4 & 4 \\ 10 & 8 \end{pmatrix}

Related Tools

Frequently Asked Questions

What is a Determinant?

The determinant is a scalar value that can be computed from the elements of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. For example, a matrix is invertible if and only if its determinant is non-zero.

When is a matrix invertible?

A square matrix is invertible (non-singular) if and only if its determinant is not equal to zero. If the determinant is zero, the matrix is singular and does not have an inverse.

How do I ensure matrix multiplication works?

To multiply two matrices A (m x n) and B (p x q), the number of columns in A (n) must equal the number of rows in B (p). The resulting matrix will have dimensions m x q.