Week 3

Question 1

Define the transpose of a matrix A.

Answer 1

Let A = (aij), m x n matrix. The transpose of A is the n x m matrix A^T := (aji).

Question 2

Define a symmetric matrix.

Answer 2

An n x n matrix A is called symmetric if A = A^T

Question 3

Define double transposition rule.

Answer 3

For any matrix A: (A^T)^T = A

Question 4

Theorem 46. Transpose of product and product of transpose.

Answer 4

Let A be a m x n matrix and B be a n x p matrix. Then (AB)^T = A^T * B^T

Question 5

What is a graph? How is it recorded?

Answer 5

A graph is a collections of vertices some pairs of which are connected by edges.
A 0-1 adjacency matrix.

Question 6

Three properties of the well defined matrices A*A^T and A^T * A where A is an m x n matrix and A^T is an n x m matrix and both are well defined.

Answer 6

They are both:
-square
-often different sizes unless matrix A is square
-are symmetric

Question 7

Define a square matrix.

Answer 7

A matrix is called square if its size is n x n for some positive integer n. The set of n x n matrices with real entries is denoted Mn(R).

Question 8

What is the main property of all n x n matrices in the set Mn(R)?

Answer 8

The set Mn(R) of n x n matrices with real coefficients is closed with respect to matrix addition and multiplication. That is, if A, B are elements of Mn(R) then A + B, AB and BA are elements of the set Mn(R).

Question 9

Define the identity matrix. Define the identity matrix theorem.

Answer 9

The n x n identity matrix In is the matrix that has on the diagonal the entries 1 and 0 elsewhere.
A * In = In * A = A

Question 10

Define invertibility or symmetric nature of a matrix.

Answer 10

An n × n - matrix A is invertible if there exists an n × n- matrix B
with A · B = In and B · A = In. The matrix B is called the inverse and instead of B we shall write A^-1.

Question 11

If A is singular is it invertible?

Answer 11

No. Singular = non-invertible. Non-singular = invertible.

Question 12

Let A be an n x n matrix. What can be said about A if it is invertible?

Answer 12

It’s inverse is unique.

Question 13

How can we check if a matrix is invertible? (Formula) What is the formula for the inverse of a 2 x 2 matrix?

Answer 13

determinant A := ad - bc != 0

A^-1 = 1 / (determinant A) * adj(A)

Question 14

If A is a 2 x 2 matrix = (a b), what is adj(A)?
(c d)

Answer 14

the adj(A) = (d -b)
(-c a)

Question 15

Let A, B be an element of Mn(R) (square matrix) and both are invertible? What can be said about AB and (AB)^-1?

Answer 15

If A and B are invertible than AB is invertible?
If A and B are invertible than (AB)^-1 = B^-1 * A^-1

Question 16

What does the following symbol mean ⇒⇐?

Answer 16

Contradiction.