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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · inverse of [[1,2],[3,4]]

Linear-Algebra · Matrix-Inverse

Inverse of [[1,2],[3,4]]

\left[\begin{matrix}1 & 2\\3 & 4\end{matrix}\right]

Àwọn ìṣàmúlò-ètò

\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]

Fi àwọn àgbèwọlé hàn

A^{-1} = \left[\begin{matrix}-2 & 1\\\frac{3}{2} & - \frac{1}{2}\end{matrix}\right]

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Bii o ṣe le: 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.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Linear Algebra

Determinants Eigenvalues and eigenvectors Matrix multiplication Row reduction Vector spaces, span and linear independence Orthogonality, projections and least squares Diagonalisation and matrix powers Linear transformations and change of basis