A3 by 3 matrix bis known to have eigenvalues 0, 1, 2. this information is enough to find three of these (give the answers where possible) : (a) the rank of b (b) thedeterminantofbtb (c) theeigenvaluesofbtb (d) the eigenvalues of (b2 + j)-1.

Answer with Step-by-step explanation:

We are given that a matrix B .

The eigenvalues of matrix are 0, 1 and 2.

a.We know that

Rank of matrix B=Number of different eigenvalues

We have three different eigenvalues

Therefore, rank of matrix B=3

b.

We know that

Determinant of matrix= Product of eigenvalues

Product of eigenvalues=

After transpose , the value of determinant remain same.

c.Let  

B=

Transpose of matrix:Rows change into columns or columns change into rows.

After transpose of matrix B

Hence, the eigenvalues of are 0, 1 and 4.

d.Eigenvalue of Identity matrix are 1, 1 and 1.

Eigenvalues of

We know that if eigenvalue of A is

Then , the eigenvalue of is

Therefore, the eigenvalues of are  

The eigenvalues of are 1, and

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