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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 · 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]

قدم ب قدم

  1. \det A = -2

    A matrix is invertible only when its determinant is non-zero.

  2. 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.

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

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

اب آپ مسئلہ منتخب کريں يا ٹائپ کريں يا خود کريں ہر قدم ، تصوير ، جواب پوشيدہ هے جب تک آپ سوال نہ کريں

آپ کے اپنے کام رکھیں

مفت اکاؤنٹ تمام درسوں پر نوٹ شامل کر تا هے ، آپ نے جو ختم کر يا هے اس کا ريکورڈ ، آپ کے حل کيے گئے مسئلے ایک جگہ پر اور آپ اس صفحے کے بارے ميں سوال کر سکتے هيں تو تيرچر ريتھ ميں خود تمام لوگوں کے ليے کھلا هے ، خواہ آپ داخل هے يا نہيں

رجسٹر لاگ ان

علامتیں

تمام تعريف کے ليے ، تصوير کے ليے اور اس ميں ہر حروف کا مطلب کے ليے کو ئي علامت کو ٹک کريں

کیسے: Matrix inverse

  1. A matrix is invertible only when its determinant is non-zero.
  2. For 2×2: swap the diagonal, negate the off-diagonal, divide by the determinant.

سوالات لوگ پوچھتے ہیں

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.

میں زیادہ Linear Algebra

DeterminantsEigenvalues and eigenvectorsMatrix multiplicationRow reductionVector spaces, span and linear independenceOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis