Matrices

Question 1

Name:

Answer 1

3x4

Question 2

What is the leading diagonal of a matrix?

Answer 2

The diagonal of numbers starting in the top left corner and finishing in the bottom right corner.

Question 3

What is an upper triangular matrix?

Answer 3

A square matrix with 0’s below the leading diagonal.

It is often denoted U.

Question 4

What is a lower triangular matrix?

Answer 4

A square matrix where all the elements above the leading diagonal are 0.

This is denoted L.

Question 5

What is a diagonal matrix?

Answer 5

A square matrix where all non-zero elements are along the leading diagonal.

It is denoted D.

Question 6

What is the trace of a matrix?

Answer 6

The sum of all the elements in the leading diagonal.

Question 7

What is the unit matrix/Identity matrix?

Answer 7

A diagonal matrix where all the elements in the leading diagonal are 1.

It is often denoted I_n or I.

Question 8

What is a zero matrix/null matrix?

Answer 8

A matrix where all the elements are 0.

It is denoted as 0 no matter the size.

Question 9

How do you transpose a matrix?

Answer 9

The rows become the columns.

It is denoted A^T .

Question 10

What makes a matrix symmetric?

Answer 10

If the transpose of the matrix is equal to the original matrix.

A = A^T

Question 11

Under what circumstances and matrices be added/subtracted?

Answer 11

If they are the same size.

Question 12

How is matrix addition commutative?

Answer 12

A+B = B+A

Question 13

How is matrix addition associative?

Answer 13

A + (B + C) = (A + B) + C

Question 14

How does the distribution law hold when adding/subtracting matrices?

Answer 14

k ( A + B ) = kA + kB

Question 15

What are the three properties of a transposed matrix?

Answer 15

( A + B )^T = A^T + B^T

( A - B )^T = A^T - B^T

( A^T )^T = A

Question 16

What is significant about the I₂ identity matrix?

Answer 16

It acts in the same way in matrix multiplication as the number 1 acts in number multiplication.

Question 17

What size matrix do you get if you multiply n x p by p x m

Answer 17

n x m

Question 18

What is the case is a matrix is singular?

Answer 18

It does not have an inverse

Question 19

9What facts can make finding determinants easier? (3)

Answer 19

• If two rows are the same or are multiples of each other the determinant is 0
• The transpose of a matrix has the same determinant as the original
• For an upper/low/diagonal matrix, the determinant is the sum of the leading diagonal

Question 20

What are the properties of eigenvalues/eigenvectors?

Answer 20

• Sum of values = trace(A)
• Product of values = |A|
• eigenvectors that correspond to distinct eigenvalues are linearly independent

Question 21

What is the modal matrix, P, of A?

Answer 21

The columns are the eigenvectors of the matrix A

Question 22

What is the fact about modal matrices?

Answer 22

P^-1AP = D

This is a similarity transformation.

Question 23

How do you raise a matrix by a power?

Answer 23

Use the fact that P^-1AP = D

Make A the subject

PP^-1APP^-1 = PDP^-1

PP^-1 = I

IAI = PDP^-1

A = PDP^-1

A^k = PD^kP^-1

For D^k each element of the diagonal is raised to the power of k

Question 24

When can an nxn matrix be diagonalised?

Answer 24

If it has n linearly independent eigenvectors

Question 25

How do you diagonalise a matrix?

Answer 25