help ii1x 3 matrix a desCA a the reduced res brs rankANA as

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i)-#i1x 3 matrix a, desCA)... a. the reduced res brs rank(A)---------N(A) as (3) decG3 5)--vs . (3) dee(036),is . (4) Am(3 ). (4) Aus ( 1 ), the eigenvalues of A are-- adj(A)

Solution

Q1) If det(A) = a, then det(cA) = c^n * a

where n represents the dimension of square matrix

Given det(A) = 3

det(5A) = 5^3 * 3 = 125 * 3 = 375

Hence the det(5A) will be equal to 375

Q2)

Rank of the matrix is equal to 3, since there are three independent columns and Null space will be equal to 1

Since the rank(A) + Null(A) = number of columns

Q3)

Expanding th determinant across the first row we get

det(A) = 1(3*5-0) + 2(0*6-0*5) + 3(0*0-3*0)

det(A) = 1(15-0) + 2(0-0) + 3(0-0)

det(A) = 15

Q4)

Det(A - pI) = 0

(2-p)(1-p) - 0 = 0

(2-p)(1-p) = 0

Hence the eigen values are 1 and 2

adjoint of the matrix will be [1,-2;0,2]