Question #297149

Let A=[a b c         ←Matrix

d e f

g h i]


where a, b, c, d,e, f, g, h, i are some real numbers, if det(A)=5 answer the following questions:


  1. Is A invertible? (Justify your answer). Find rank(A)


  1. Let b= [a+d b+e c+f         ←Matrix

d      e       f

2g     2h    2i]


And, C= [a b c                   ←Matrix

     -2d -2e -2f

     3g 3h 3I ]


Compute det(B) and det(C).



(C) Compute det(A^-1) and det(adj(A)).




1
Expert's answer
2022-02-14T16:07:22-0500

a)


A is invertible because it's determinant is not zero.

Rank of A =3


b)

Det B

Row 2 of A is added to get row 1 of B and twice row 3 of A is row 3 of B.

\therefore det B =2×5=2×5

=10=10



Det C

(-Twice) row 2 of A and 3 times row 3 of A is row 2 and row 3 of C.

\therefore det C =(2)×(3)×(5)=(-2)×(3)×(5)

=30=-30


c)

Det (A) det (( A1)=1^{-1})=1

    \implies det (A1)=15(A^{-1})=\frac{1}{5}


Det (adj A) == det (A)n1(A)^{n-1}

=(5)31=(5)^{3-1}

=25=25





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