Matrix inverse: Difference between revisions
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=== Calculation === | === Calculation === | ||
The inverse may be computed from the [[adjugate matrix]], which shows that a matrix is invertible if and only if its [[determinant]] is itself invertible: over a [[field (mathematics)|field]] such as the [[real number|real]] or [[complex number]]s, this is equivalent to specifying that the determinant does not equal zero. | The inverse may be computed from the [[adjugate matrix]], which shows that a matrix is invertible if and only if its [[determinant]] is itself invertible: over a [[field (mathematics)|field]] such as the [[real number|real]] or [[complex number]]s, this is equivalent to specifying that the determinant does not equal zero.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:01, 16 September 2024
In matrix algebra, the inverse of a square matrix A is X if
where In is the n-by-n identity matrix
If this equation is true, X is the inverse of A, denoted by A-1. A is also the inverse of X.
A matrix is invertible if and only if it possesses an inverse.
Uniqueness
Every invertible matrix has only one inverse.
For example, if AX = I and AY = I, then X = Y. So, X = Y = A-1.
To prove this, consider the case of X.A.Y.
Calculation
The inverse may be computed from the adjugate matrix, which shows that a matrix is invertible if and only if its determinant is itself invertible: over a field such as the real or complex numbers, this is equivalent to specifying that the determinant does not equal zero.