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In linear algebra, the study of matrices is one of the fundamental basis of this subject. One of the concepts within this study, is the notion of an invertible or nonsingular matrix.
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In linear algebra, the study of matrices is one of the fundamental basis of this subject. One of the concepts within this study, is the notion of an invertible or nonsingular matrix.  
 +
 
 +
 
 +
 
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== Definition ==
 +
 
 +
A square matrix is said to be '''invertible '''or '''nonsingular''', if when multiplied by another similar matrix, the result yields the identity matrix.
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 +
 
 +
 
 +
Let A and B be n × n matrices.
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 +
 
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A is invertible or nonsingular and B is its inverse if:
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<math>% MathType!MTEF!2!1!+-
 +
% feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
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% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
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% GaamOqaiabg2da9iaadkeacaWGbbGaeyypa0JaamysamaaBaaaleaa
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% caWGUbaabeaaaOqaaaqaamaayeaabaWaaeWaaeaafaqabeabeaaaaa
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% qaaiaadggadaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaamyyamaa
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% baaSqaaiaaigdacaWGUbaabeaaaOqaaiaadggadaWgaaWcbaGaaGOm
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% aiaaigdaaeqaaaGcbaGaamyyamaaBaaaleaacaaIYaGaaGOmaaqaba
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% aakeaacqWIVlctaeaacaWGHbWaaSbaaSqaaiaaikdacaWGUbaabeaa
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% aOqaaiabl6Uinbqaaiabl6UinbqaaiablgVipbqaaiabl6Uinbqaai
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% aadggadaWgaaWcbaGaamOBaiaaigdaaeqaaaGcbaGaamyyamaaBaaa
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% baaakeaacaWGIbWaaSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiabl+
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% UimbqaaiaadkgadaWgaaWcbaGaaGOmaiaad6gaaeqaaaGcbaGaeSO7
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% I0eabaGaeSO7I0eabaGaeSy8I8eabaGaeSO7I0eabaGaamOyamaaBa
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% aaleaacaWGUbGaaGymaaqabaaakeaacaWGIbWaaSbaaSqaaiaad6ga
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% caaIYaaabeaaaOqaaiabl+UimbqaaiaadkgadaWgaaWcbaGaamOBai
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% aad6gaaeqaaaaaaOGaayjkaiaawMcaaaWcbaGaamOqaaGccaGL34pa
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% cqGH9aqpdaagbaqaamaabmaabaqbaeqabqabaaaaaeaacaaIXaaaba
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% GaaGimaaqaaiabl+UimbqaaiaaicdaaeaacaaIWaaabaGaaGymaaqa
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% aiabl+UimbqaaiaaicdaaeaacqWIUlstaeaacqWIUlstaeaacqWIXl
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% YtaeaacqWIUlstaeaacaaIWaaabaGaaGimaaqaaiabl+Uimbqaaiaa
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% igdaaaaacaGLOaGaayzkaaaaleaacaWGjbWaaSbaaWqaaiaad6gaae
 +
% qaaaGccaGL34paaaaa!ACB0!
 +
$$\eqalign{
 +
  & AB = BA = {I_n}  \cr
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  &  \cr
 +
  & \overbrace {\left( {\matrix{
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  {{a_{11}}} & {{a_{12}}} &  \cdots  & {{a_{1n}}}  \cr
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  {{a_{21}}} & {{a_{22}}} &  \cdots  & {{a_{2n}}}  \cr
 +
    \vdots  &  \vdots  &  \ddots  &  \vdots  \cr
 +
  {{a_{n1}}} & {{a_{n2}}} &  \cdots  & {{a_{nn}}}  \cr
 +
 
 +
} } \right)}^A\overbrace {\left( {\matrix{
 +
  {{b_{11}}} & {{b_{12}}} &  \cdots  & {{b_{1n}}}  \cr
 +
  {{b_{21}}} & {{b_{22}}} &  \cdots  & {{b_{2n}}}  \cr
 +
    \vdots  &  \vdots  &  \ddots  &  \vdots  \cr
 +
  {{b_{n1}}} & {{b_{n2}}} &  \cdots  & {{b_{nn}}}  \cr
 +
 
 +
} } \right)}^B = \overbrace {\left( {\matrix{
 +
  1 & 0 &  \cdots  & 0  \cr
 +
  0 & 1 &  \cdots  & 0  \cr
 +
    \vdots  &  \vdots  &  \ddots  &  \vdots  \cr
 +
  0 & 0 &  \cdots  & 1  \cr
 +
 
 +
} } \right)}^{{I_n}} \cr} $$</math>
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[[Category:MA265Fall2010Walther]]
 
[[Category:MA265Fall2010Walther]]

Revision as of 15:17, 27 November 2010

The Inverse of a Matrix

In linear algebra, the study of matrices is one of the fundamental basis of this subject. One of the concepts within this study, is the notion of an invertible or nonsingular matrix.


Definition

A square matrix is said to be invertible or nonsingular, if when multiplied by another similar matrix, the result yields the identity matrix.


Let A and B be n × n matrices.


A is invertible or nonsingular and B is its inverse if:


$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGbb % GaamOqaiabg2da9iaadkeacaWGbbGaeyypa0JaamysamaaBaaaleaa % caWGUbaabeaaaOqaaaqaamaayeaabaWaaeWaaeaafaqabeabeaaaaa % qaaiaadggadaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaamyyamaa % BaaaleaacaaIXaGaaGOmaaqabaaakeaacqWIVlctaeaacaWGHbWaaS % baaSqaaiaaigdacaWGUbaabeaaaOqaaiaadggadaWgaaWcbaGaaGOm % aiaaigdaaeqaaaGcbaGaamyyamaaBaaaleaacaaIYaGaaGOmaaqaba % aakeaacqWIVlctaeaacaWGHbWaaSbaaSqaaiaaikdacaWGUbaabeaa % aOqaaiabl6Uinbqaaiabl6UinbqaaiablgVipbqaaiabl6Uinbqaai % aadggadaWgaaWcbaGaamOBaiaaigdaaeqaaaGcbaGaamyyamaaBaaa % leaacaWGUbGaaGOmaaqabaaakeaacqWIVlctaeaacaWGHbWaaSbaaS % qaaiaad6gacaWGUbaabeaaaaaakiaawIcacaGLPaaaaSqaaiaadgea % aOGaay5n+dWaaGraaeaadaqadaqaauaabeqaeqaaaaaabaGaamOyam % aaBaaaleaacaaIXaGaaGymaaqabaaakeaacaWGIbWaaSbaaSqaaiaa % igdacaaIYaaabeaaaOqaaiabl+UimbqaaiaadkgadaWgaaWcbaGaaG % ymaiaad6gaaeqaaaGcbaGaamOyamaaBaaaleaacaaIYaGaaGymaaqa % baaakeaacaWGIbWaaSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiabl+ % UimbqaaiaadkgadaWgaaWcbaGaaGOmaiaad6gaaeqaaaGcbaGaeSO7 % I0eabaGaeSO7I0eabaGaeSy8I8eabaGaeSO7I0eabaGaamOyamaaBa % aaleaacaWGUbGaaGymaaqabaaakeaacaWGIbWaaSbaaSqaaiaad6ga % caaIYaaabeaaaOqaaiabl+UimbqaaiaadkgadaWgaaWcbaGaamOBai % aad6gaaeqaaaaaaOGaayjkaiaawMcaaaWcbaGaamOqaaGccaGL34pa % cqGH9aqpdaagbaqaamaabmaabaqbaeqabqabaaaaaeaacaaIXaaaba % GaaGimaaqaaiabl+UimbqaaiaaicdaaeaacaaIWaaabaGaaGymaaqa % aiabl+UimbqaaiaaicdaaeaacqWIUlstaeaacqWIUlstaeaacqWIXl % YtaeaacqWIUlstaeaacaaIWaaabaGaaGimaaqaaiabl+Uimbqaaiaa % igdaaaaacaGLOaGaayzkaaaaleaacaWGjbWaaSbaaWqaaiaad6gaae % qaaaGccaGL34paaaaa!ACB0! $$\eqalign{ & AB = BA = {I_n} \cr & \cr & \overbrace {\left( {\matrix{ {{a_{11}}} & {{a_{12}}} & \cdots & {{a_{1n}}} \cr {{a_{21}}} & {{a_{22}}} & \cdots & {{a_{2n}}} \cr \vdots & \vdots & \ddots & \vdots \cr {{a_{n1}}} & {{a_{n2}}} & \cdots & {{a_{nn}}} \cr } } \right)}^A\overbrace {\left( {\matrix{ {{b_{11}}} & {{b_{12}}} & \cdots & {{b_{1n}}} \cr {{b_{21}}} & {{b_{22}}} & \cdots & {{b_{2n}}} \cr \vdots & \vdots & \ddots & \vdots \cr {{b_{n1}}} & {{b_{n2}}} & \cdots & {{b_{nn}}} \cr } } \right)}^B = \overbrace {\left( {\matrix{ 1 & 0 & \cdots & 0 \cr 0 & 1 & \cdots & 0 \cr \vdots & \vdots & \ddots & \vdots \cr 0 & 0 & \cdots & 1 \cr } } \right)}^{{I_n}} \cr} $$ $

Alumni Liaison

Prof. Math. Ohio State and Associate Dean
Outstanding Alumnus Purdue Math 2008

Jeff McNeal