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Question

Show that A=[5    31  2] satisfies the equation A2 - 3A - 7I = 0 and hence find the value  of A1


Solution

We have  A=[5    31  2]

A2 = A. A =[5    31  2][5    31  2]

[253   1565+2   3+4]=[22    93   1]

3A = 3  [5  31  2]=[15  93  6]

and   7I = 7[1  00  1]=[7  00  7]

A23A7I=[22  93  1][15  93  6][7  00  7]

=[22157  9903+30   1+6+7]

=[0   00   0]

=0

Since,  A2 - 3A - 7I =0

A1[(A2)3A7I]=A10

A1A.A3A1 A7 A1 I=0         [A10=0]

IA3I7A1=0            [A1A=I]

A3I7A1=0            [A1 I=A1]

7A1=A+3I

[5  31     2]+[3    00   3]=[2  31   5]

A1=17[2  31   5]

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