Week 2

Question 1

What is a matrix?

Answer 1

A matrix is a rectangular array of numbers called the entries or elements of the matrix.

Question 2

What is the transpose of a matrix?

Answer 2

The transpose of a mxn matrix A is the nxm matrix A^T obtained by interchanging the rows and columns of A.

Question 3

What is a symmetric matrix?

Answer 3

A square matrix is symmetric if A^T=A. That is, A is equal to its own transpose.

Question 4

What is linear combination for matrices?

Answer 4

If A1,…,An are mxn matrices and c1,…,ck are scalars, we may form the linear combination
c1A1+c2A2+…+ckAk

Question 5

What is a span of a matrix?

Answer 5

If S={A1,…,Ak} is a set of mxn matrices, then the span of S is the set of all linear combinations of the elements of S. That is,
Span S= {c1A1+…+ckAk: c1,…,ck are scalars}

Question 6

What are linear independent matrices?

Answer 6

A collection {A1,…,Ak} of mxn matrices is linearly independent if the only solution to the equation
c1A1+c2A2+…+ckAk=0
for scalars c1,…,ck is c1=c2=…=ck=0. If there are non trivial coefficients which satisfy this equation then the set {A1,…,Ak} is linearly dependent.

Question 7

What is the inverse of a matrix?

Answer 7

If A is an nxn matrix, the inverse of A is an nxn matrix A’ such that
AA’=In & A’A=In
If A’ exists we say A is invertible. If no inverse exists, we say A is not invertible.

Question 8

What is theorem 3.6?

Answer 8

If an nxn matrix A is invertible then its inverse is unique

Question 9

What is theorem 3.7?

Answer 9

If A is an invertible nxn matrix then the system of linear equations given by Ax=b has the unique solution given by x=A^-1b

Question 10

What is theorem 3.8?

Answer 10

If A= [a b]
         [c d]
Then A is invertible if ad-bc≠0, in which case
A-1= 1/(ad-bc) [d -b]
                          [-c a]
If ad-bc=0 then A is not invertible.

Question 11

What is the determinant?

Answer 11

For a 2x2 matrix
A= [a b]
      [c d]
We call ab-bc the determinant so that
ad-bc=0