🔢 Linear Algebra & Matrix Analysis
Matrix Determinant & Inverse Calculator
Compute matrix determinants ($\det(A)$), inverse matrices ($A^{-1}$), trace ($\operatorname{tr}(A)$), and eigenvalues for $2\times 2$ and $3\times 3$ matrices.
Matrix Elements A
Matrix Determinant det(A)
det(A) = 10.00
Inverse Matrix $A^{-1}$ ($A \cdot A^{-1} = I$):
[ 0.600, -0.700 ]
[ -0.200, 0.400 ]
[ -0.200, 0.400 ]
Matrix Trace ($\operatorname{tr}(A)$)
10.00
Eigenvalues ($\lambda_1, \lambda_2$)
8.828, 1.172
💡 Determinant Formula:
$$\det\begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix} = (4)(6) - (7)(2) = 24 - 14 = 10$$
Matrix Inversion Formulas
For a $2\times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$:
- Determinant: $$\det(A) = ad - bc$$
- Inverse: $$A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$
- Eigenvalues: Roots of the characteristic polynomial $\lambda^2 - \operatorname{tr}(A)\lambda + \det(A) = 0$.