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Matrix inverse
The 2×2 formula and Gauss-Jordan elimination for the rest.
A⁻¹ is the matrix that undoes A. For 2×2 there is a formula; for anything bigger, row-reduce [A | I] until A becomes I, and the identity becomes A⁻¹. Every row operation is shown.
مثال عملي · inverse of [[1,2],[3,4]]
Inverse of [[1,2],[3,4]]
خطوة خطوة
- \det A = -2
A matrix is invertible only when its determinant is non-zero.
- A^{-1} = \frac{1}{\det A}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \frac{1}{-2}\left[\begin{matrix}4 & -2\\-3 & 1\end{matrix}\right]
For 2×2: swap the diagonal, negate the off-diagonal, divide by the determinant.
- A^{-1} = \left[\begin{matrix}-2 & 1\\\frac{3}{2} & - \frac{1}{2}\end{matrix}\right]
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كيف: Matrix inverse
- A matrix is invertible only when its determinant is non-zero.
- For 2×2: swap the diagonal, negate the off-diagonal, divide by the determinant.
أسئلة يطرحها الناس
What does a determinant mean geometrically?
It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.
What is an eigenvector?
A direction the matrix does not turn: it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.
Why is matrix multiplication not commutative?
Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.
أكثر في Linear Algebra
DeterminantsEigenvalues and eigenvectorsMatrix multiplicationRow reductionVector spaces, span and linear independenceOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis