Matrices & Vectors Solver
Inverse Matrix Calculator & Matrix Operations Solver
A matrix is just a grid of numbers, but it has rules of its own. This solver works as an inverse matrix calculator, determinant calculator, matrix multiplication tool and Cramer's rule solver. Enter a 2×2 or 3×3 matrix in LaTeX form and the engine returns the inverse, determinant or product with step-by-step working.
How to use the inverse matrix calculator
To find A⁻¹, enter your matrix using the bracket notation. For a 2×2 matrix:
- Type
\begin{pmatrix}2&1\\3&4\end{pmatrix}^{-1}to get the inverse - For determinant, type
\det\begin{pmatrix}2&1\\3&4\end{pmatrix} - For multiplication:
\begin{pmatrix}1&2\\3&4\end{pmatrix} \cdot \begin{pmatrix}5&6\\7&8\end{pmatrix}
The inverse formula (2×2 quick reference)
For a 2×2 matrix $A = \begin{pmatrix}a&b\\c&d\end{pmatrix}$, the inverse is:
Example: for $A = \begin{pmatrix}2&1\\3&4\end{pmatrix}$, $\det(A) = 2 \cdot 4 - 1 \cdot 3 = 5$. So $A^{-1} = \dfrac{1}{5}\begin{pmatrix}4&-1\\-3&2\end{pmatrix} = \begin{pmatrix}0.8&-0.2\\-0.6&0.4\end{pmatrix}$.
The inverse formula (3×3 via adjoint method)
For 3×3 matrices, use the adjoint method: $A^{-1} = \dfrac{\text{adj}(A)}{\det(A)}$. The adjoint is the transpose of the cofactor matrix. The cofactor of each element involves a 2×2 minor determinant with the right sign (the +/− checkerboard pattern). This is the method CBSE Class 12 board examiners want to see.
When does a matrix have no inverse?
A matrix is singular (non-invertible) when its determinant is zero. Geometrically, the matrix collapses space into a lower dimension — it can't be reversed. In practice, det(A) = 0 usually means the rows or columns are linearly dependent (one is a multiple or sum of others).
Common matrix operations covered
$\det = ad - bc$
$a(ei - fh) - b(di - fg) + c(dh - eg)$
$(AB)_{ij} = \sum_k A_{ik} B_{kj}$
$x_i = \det(A_i) / \det(A)$
$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$
$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$