Show that the matrix A = [ * 20 c 2&3 1&2 ] satisfies the equation A^2 – 4A + I = 0. where I is 2 2 identity matrix and O is 2 2 zero matrix. Using this equation, find A^ -1

A^2 = [ * 20 c 2&3 1&2 ]. [ * 20 c 2&3 1&2 ]= [ * 20 c 7& 12 1&7 ] A^2 - 4A + I = [ * 20 c 7& 12 1&7 ] - [ * 20 c 8& 12 4&8 ] + [ * 20 c 1&0 0&1 ] = [ * 20 c 0&0 0&0 ] = 0 A^2 - 4A + I = 0 A^2 - 4A = - I AA A^ - 1 - 4A A^ - 1 = I A^ - 1 AI - 4I = - I A^ - 1 [ A A^ - 1 = I ] A^ - 1 = 4I - A = [ * 20 c 2& - 3 - 1 &2 ]

Show that the matrix A = [ * 20 c 2&3 1&2 ] satisfies the equatio…

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Question

Show that the matrix $A = \left[ {\begin{array}{*{20}{c}} 2&3 \\ 1&2 \end{array}} \right]$ satisfies the equation $A^2 – 4A + I = 0.$ where I is $2 \times 2$ identity matrix and $O$ is $2 \times 2$ zero matrix. Using this equation, find $A^{-1}$

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Answer

${A^2} = \left[ {\begin{array}{*{20}{c}} 2&3 \\ 1&2 \end{array}} \right].\left[ {\begin{array}{*{20}{c}} 2&3 \\ 1&2 \end{array}} \right]$$= \left[ {\begin{array}{*{20}{c}} 7&{12} \\ 1&7 \end{array}} \right]$
${A^2} - 4A + I = \left[ {\begin{array}{*{20}{c}} 7&{12} \\ 1&7 \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} 8&{12} \\ 4&8 \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right]$
$= \left[ {\begin{array}{*{20}{c}} 0&0 \\ 0&0 \end{array}} \right]$
$= 0$
${A^2} - 4A + I = 0$
${A^2} - 4A = - I$
$AA{A^{ - 1}} - 4A{A^{ - 1}} = I{A^{ - 1}}$
$AI - 4I = - I{A^{ - 1}}\left[ {\because A{A^{ - 1}} = I} \right]$
${A^{ - 1}} = 4I - A$
$= \left[ {\begin{array}{*{20}{c}} 2&{ - 3} \\ { - 1}&2 \end{array}} \right]$