l Verify A( adj. A) =(adj . A) A=|A| : [ ccc 1 & -1 & 2 3 & 0 & -2 1 & 0 & 3 ]

Let A = [ * 20 c 1& - 1 &2 3&0& - 2 1&0&3 ] | A | = [ * 20 c 1& - 1 &2 3&0& - 2 1&0&3 ] A_ 11 =+ | * 20 c 0& - 2 0&3 | = + 0 + 0 = 0, A_ 12 = - | * 20 c 3& - 2 1&3 | = - ( 9 + 2 ) = - 11 A_ 13 = + | * 20 c 3&0 1&0 | = + ( 0 - 0 ) = 0, A_ 21 = - | * 20 c - 1 &2 0&3 | - ( - 3 - 0 ) = 3 A_ 22 = + | * 20 c 1&2 1&3 | = 3 - 2 = 1, A_ 23 = - | * 20 c 1& - 1 1&0 | = - ( 0 + 1 ) = - 1 A_ 31 = + | * 20 c - 1 &2 0& - 2 | = 2 - 0 = 2, A_ 32 = - | * 20 c 1&2 3& - 2 | = - ( - 2 - 6 ) = 8 A_ 33 = + | * 20 c 1& - 1 3&0 | = 3 + 0 = 3 adj. A = | * 20 c 0& - 11 &0 3&1& - 1 2&8&3 | = | * 20 c 0&3&2 - 11 &1&8 0& - 1 &3 | A. ( adj.A ) = [ * 20 c 1& - 1 &2 3&0& - 2 1&0&3 ] [ * 20 c 0&3&2 - 11 &1&8 0& - 1 &3 ] [ * 20 c 0 + 11 + 0 & 3 - 1 - 2 & 2 - 8 + 6 0 - 0 - 0 & 9 + 0 + 2 & 6 + 0 - 6 0 + 0 + 0 & 3 + 0 - 3 & 2 + 0 + 9 ] = [ * 20 c 11 &0&0 0& 11 &0 0&0& 11 ]...(i) Again (adj. A). A= [ * 20 c 0&3&2 - 11 &1&8 0& - 1 &3 ] [ * 20 c 1& - 1 &2 3&0& - 2 1&0&3 ] = [ * 20 c 0 + 9 + 2 & 0 + 0 + 0 & 0 - 6 + 6 - 11 + 3 + 8 & 11 + 0 + 0 & - 22 - 2 + 24 0 - 3 + 3 & 0 - 0 + 0 & 0 + 2 + 9 ] = [ * 20 c 11 &0&0 0& 11 &0 0&0& 11 ]….(ii) And | A | = | * 20 c 1& - 1 &2 3&0& - 2 1&0&3 | = 1 ( 0 - 0 ) - ( - 1 ) ( 9 + 2 ) + 2 ( 0 - 0 ) = 0 + 11 + 0 = 11 Also | A |I = | A | I_3 = 11 [ * 20 c 1&0&0 0&1&0 0&0&1 ] = [ * 20 c 11 &0&0 0& 11 &0 0&0& 11 ]...(iii) From eq. (i), (ii) and (iii) A. (adj. A) = (adj. A). A = |A|

l Verify A( adj. A) =(adj . A) A=|A| : [ ccc 1 & -1 & 2 3 & 0 & -…

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Question

$\begin{array}{l}\text { Verify } A(\text { adj. A) }=(\operatorname{adj} . A) A=|A| \text { : } \\ {\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3\end{array}\right]}\end{array}$

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Answer

Let $A = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&2 \\ 3&0&{ - 2} \\ 1&0&3 \end{array}} \right]$
$\Rightarrow \left| A \right| = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&2 \\ 3&0&{ - 2} \\ 1&0&3 \end{array}} \right]$
$\therefore {A_{11}} =+ \left| {\begin{array}{*{20}{c}} 0&{ - 2} \\ 0&3 \end{array}} \right| = + 0 + 0 = 0,$
${A_{12}} = - \left| {\begin{array}{*{20}{c}} 3&{ - 2} \\ 1&3 \end{array}} \right| = - \left( {9 + 2} \right) = - 11$
${A_{13}} = + \left| {\begin{array}{*{20}{c}} 3&0 \\ 1&0 \end{array}} \right| = + \left( {0 - 0} \right) = 0,$
${A_{21}} = - \left| {\begin{array}{*{20}{c}} { - 1}&2 \\ 0&3 \end{array}} \right| - \left( { - 3 - 0} \right) = 3$
${A_{22}} = + \left| {\begin{array}{*{20}{c}} 1&2 \\ 1&3 \end{array}} \right| = 3 - 2 = 1,$
${A_{23}} = - \left| {\begin{array}{*{20}{c}} 1&{ - 1} \\ 1&0 \end{array}} \right| = - \left( {0 + 1} \right) = - 1$
${A_{31}} = + \left| {\begin{array}{*{20}{c}} { - 1}&2 \\ 0&{ - 2} \end{array}} \right| = 2 - 0 = 2,$
${A_{32}} = - \left| {\begin{array}{*{20}{c}} 1&2 \\ 3&{ - 2} \end{array}} \right| = - \left( { - 2 - 6} \right) = 8$
${A_{33}} = + \left| {\begin{array}{*{20}{c}} 1&{ - 1} \\ 3&0 \end{array}} \right| = 3 + 0 = 3$
$\therefore adj.\ A = \left| {\begin{array}{*{20}{c}} 0&{ - 11}&0 \\ 3&1&{ - 1} \\ 2&8&3 \end{array}} \right|$
$= \left| {\begin{array}{*{20}{c}} 0&3&2 \\ { - 11}&1&8 \\ 0&{ - 1}&3 \end{array}} \right|$
$\therefore A.\left( {adj.A} \right) = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&2 \\ 3&0&{ - 2} \\ 1&0&3 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0&3&2 \\ { - 11}&1&8 \\ 0&{ - 1}&3 \end{array}} \right]$
$\left[ {\begin{array}{*{20}{c}} {0 + 11 + 0}&{3 - 1 - 2}&{2 - 8 + 6} \\ {0 - 0 - 0}&{9 + 0 + 2}&{6 + 0 - 6} \\ {0 + 0 + 0}&{3 + 0 - 3}&{2 + 0 + 9} \end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}} {11}&0&0 \\ 0&{11}&0 \\ 0&0&{11} \end{array}} \right]...(i)$
Again $(adj. \ A). A= \left[ {\begin{array}{*{20}{c}} 0&3&2 \\ { - 11}&1&8 \\ 0&{ - 1}&3 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1&{ - 1}&2 \\ 3&0&{ - 2} \\ 1&0&3 \end{array}} \right]$
$= \left[ {\begin{array}{*{20}{c}} {0 + 9 + 2}&{0 + 0 + 0}&{0 - 6 + 6} \\ { - 11 + 3 + 8}&{11 + 0 + 0}&{ - 22 - 2 + 24} \\ {0 - 3 + 3}&{0 - 0 + 0}&{0 + 2 + 9} \end{array}} \right]$
$= \left[ {\begin{array}{*{20}{c}} {11}&0&0 \\ 0&{11}&0 \\ 0&0&{11} \end{array}} \right]….(ii)$
And $\left| A \right| = \left| {\begin{array}{*{20}{c}} 1&{ - 1}&2 \\ 3&0&{ - 2} \\ 1&0&3 \end{array}} \right|$
$= 1\left( {0 - 0} \right) - \left( { - 1} \right)\left( {9 + 2} \right) + 2\left( {0 - 0} \right) $
$= 0 + 11 + 0 $
$= 11$
Also $\left| A \right|I = \left| A \right|{I_3} = 11\left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {11}&0&0 \\ 0&{11}&0 \\ 0&0&{11} \end{array}} \right]...(iii)$
$\therefore$ From $eq. (i), (ii)$ and $(iii) A. (adj.\ A) = (adj. \ A). A = |A|$