Chapter 1.7

Question 0

A lower triangular matrix

Answer 0

A square matrix where all the entries above the main diagonal are zero.

Question 1

how does multiplication with diagonal matrices work

Answer 1

To multiply a matrix A on the left with a diagonal matrix D. Simply multiply each succesive row of A with the succesive diagonal entries of D. To multiply a matrix A on the right with a diagonal matrix D. Simply multiply each succesive column of A with the succesive diagonal entries of D.

Question 2

An upper diagonal matrix

Answer 2

A square matrix where all the entries below the main diagonal are zero.

Question 3

Triangular matrices (4)

Answer 3

1) The transpose of a lower triangular matrix is an upper triangular matrix and the transpose of an upper triangular mattix is a lower triangular matrix.
2) The product of two upper triangular matrices is upper triangular and the product of two lower triangular matrices is lower triangular.
3) A triangular matrix is invertible iff all the entries on the main diagonal are nonzero
4) The inverse of an invertible lower triangular matrix is lower triangular and the inverse of an invertible upper triangular matrix is upper triangular.

Question 4

Def: symmetric matrices

Answer 4

A square matrix is said to be symmetric of A = transpose(A)

Question 5

If A and B are symmetric matrices with the same size, and if k is any scalar then (3)

Answer 5

1) Transpose(A) is symmetric
2) A + B and A - B are symmetric
3) kA is symmetric

Question 6

The product of two symmetric matrices is symmetric iff

Answer 6

The matrices commute

Question 7

If A is an invertible symmetric matrix then

Answer 7

Inv(A) is symmetric

Question 8

If A is an invertible matrix then what can be said of A*transpose(A)

Answer 8

Atranspose(A) and transpose(A)A are also invertible