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Question
If `A= [[1,2,0],[3,-4,5],[0,-1,3]]` compute A2 − 4A + 3I3.
Solution
Given : A= `[[1, 2,0],[3,-4,5],[0,-1,3]]`
Now,
`A^2`=AA
`⇒A^2=` `[[1,2,0],[3,-4 ,5],[0,-1 ,3]]``[[1,2,0],[3,-4,5],[0,-1,3]]`
`⇒A^2=` `[[1+6+0,2-8-0,0+10+0],[3-12+0,6+16-5,0-20+15],[0-3+0,0+4-3,0-5+9 ]]`
`⇒A^2=` `[[7,-6,10],[-9,17,-5],[-3,1,4]]`
`A^2−4A+3I_3`
⇒`A^2−4A+3I_3`=`[[7,-6,10],[-9,17,-5],[-3,1,4]]-4``[[1,2,0],[3,-4,5],[0,-1,3]]+3[[1,0,0],[0,1,0],[0,0,1]]`
⇒`A^2−4A+3I_3`= `[[7,-6,10],[-9,17,-5],[-3,1,4]]-[[4,8,0],[12,-16,20],[0,-4,12]]+[[3,0,0],[0,3,0],[0,0,3]]`
⇒ `A^2−4A+3I_3`=`[[7-4+3 -6-8+0 10-0+0 10-0+0],[-9-12+0 17+16+3 -5-20+0],[-3-0+0 1+4+0 4-12+3]]`
⇒ `A^2−4A+3I_3`= `[[6,-14,10],[-21,36,-25],[-3,5,-5]]`
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