Question 1: Matrix by Condition

Give two example matrices for each of the following conditions. Enter each matrix in a separate answer field.

(a) Enter two 3 ⨯ 3 symmetric matrices. All off-diagonal elements cannot be the same.

[0.1] [0]

Possible answers include:

( − 2 − 1 − 8 − 1 − 6 − 2 − 8 − 2 − 4 ) and ( − 1 − 8 − 3 − 8 − 7 − 2 − 3 − 2 − 2 )

Step 1 of 2

(a) A symmetric matrix is a square matrix that is equal to its transpose. In other words, A is a symmetric matrix if it satisfies A = A T .

This means that the entry in row i , column j is the same as the entry in row j , column i :

a i⁣j = a j⁣i for all i , j

Step 2 of 2

Some examples of symmetric matrices are:

( − 2 − 1 − 8 − 1 − 6 − 2 − 8 − 2 − 4 )

( − 1 − 8 − 3 − 8 − 7 − 2 − 3 − 2 − 2 )


(b) Enter two 3 ⨯ 3 singular matrices. No row or column should be all zeros.

[0.2] [0]

Possible answers include:

( 1 1 2 3 5 4 2 2 4 ) and ( 1 1 1 1 0 1 2 1 2 )

Step 1 of 2

(b) A singular matrix A is a square matrix that has these properties:

• ❘ A ❘ = 0
• Its rows (or columns) are linearly dependent (one row or column can be written as a combination of the others).
• Its rank is less than the number of rows (or columns).
• The equation A ⁢ x = b does not have a unique solution for every b .

We can use the second property to create singular matrices.

Step 2 of 2

Create matrices that have at least one row or column linearly dependent on the rest.

For example:

R 3 = R 1 + 4 ⁢ R 2
    
( a 1 ⁢ ​ 1 a 1 ⁢ ​ 2 a 1 ⁢ ​ 3 a 2 ⁢ ​ 1 a 2 ⁢ ​ 2 a 2 ⁢ ​ 3 a 1 ⁢ ​ 1 + 4 ⁢ a 2 ⁢ ​ 1 a 1 ⁢ ​ 2 + 4 ⁢ a 2 ⁢ ​ 2 a 1 ⁢ ​ 3 + 4 ⁢ a 2 ⁢ ​ 3 )
    
Choose the first two rows and compute the third one:
    
( 0 − 5 − 10 0 1 0 − 7 0 − 6 10 − 23 − 34 41 )

Also, we can just repeat rows or columns to create singular matrices.


(c) Enter two 4 ⨯ 4 matrices with rank 3 .

[0.35] [0]

Possible answers include:

( − 7 0 − 6 − 4 − 7 − 3 − 10 − 8 − 3 − 9 0 − 9 − 0 − 9 − 1 0 − 7 − 9 − 1 ) and ( 1 0 0 1 0 1 0 0 2 0 1 2 3 0 1 3 )

Step 1 of 2

(c) The rank of a matrix is the number of linearly independent rows or columns. To create a matrix of rank 3 , we can choose 3 independent rows or columns and compute the rest as linear combinations of these.

For simplicity, the linear combinations can just be repetitions of the chosen rows or columns.

Step 2 of 2

Here are 3 independent columns:
    
( − 7 − 3 − 9 − 1 ) ​ , ​ ( 0 − 6 − 10 0 − 9 0 − 7 ) ​ , ​ ( − 4 − 8 − 0 − 9 )
    
We can simply repeat the first column for the remaining columns:
    
( − 7 0 − 6 − 4 − 7 − 3 − 10 − 8 − 3 − 9 0 − 9 − 0 − 9 − 1 0 − 7 − 9 − 1 )


(d) Enter two 3 ⨯ 3 matrices with eigenvalues { 1 , 4 , − 1 } .

[0.35] [0]

Possible answers include:

( 1 − 4 − 0 0 − 4 − 4 0 − 0 − 1 ) and ( 1 1 − 0 0 4 − 1 0 0 − 1 )

Step 1 of 1

(d) We know that the eigenvalues of any triangular matrix are exactly its diagonal entries (counting multiplicity).

We can use the above fact to create a matrix with eigenvalues { 1 , 4 , − 1 } . The diagonal elements can be these given eigenvalues in any order, and the remaining entries can be anything.

For example:

( 1 − 3 − 5 0 − 4 − 5 0 − 0 − 1 )