Homework 3

Problemset #1

(1)

Rank of the matrix A is 4.

(2)

For an m x n matrix, If m is less than n, then the maximum rank of the matrix is m. If m is greater than n, then the maximum rank of the matrix is n. If m is greater than n, the maximum rank of the matrix is n. The rank of a matrix would be zero only if the matrix had no elements. If a matrix had even one element, its minimum rank would be one.

(3)

In the matrix B, row 2 and row 3 are scalar multiple of row 1. Since rows 1, 2, 3 are linearly dependent. Matrix B only has one linearly independent row, so its rank is 1.

Problemset #2

Eigenvalues and eigenvectors of Matrix A

The eigenvalues are:

λ1=1 λ2=4 λ3=6

Vectors for λ1:

xλ=1=[1\0\0]

λ=4

Vectors for λ2:

xλ=4=[2/3\1\0]

λ=6

Vectors for λ3:

x3=1,x2=5/2,andx1=8/5

xλ=6=[8/5\5/2\1]