Chapter 2.3

Question 0

Let A, B, and C be nxn matrices of the same size then det(C) = det(A) + det(A) iff

Answer 0

They only differ in one row, say the r:th, and the r:th row of C can be obtained by adding the corresponding entries in te r:th row of A and B. The same holds for columns.

Question 1

If A is an nxn matrix then det(kA) equals

Answer 1

K^n*det(A)

Question 2

If A and B are square matrices of the same size then det(AB) equals

Answer 2

det(A) * det(B)

Question 3

A square matrix is invertible if it’s determinant

Answer 3

Does not equal zero

Question 4

If A is invertible, then the determinant is

Answer 4

det(inv(A)) = 1/det(A)

Question 5

Def: the adjoint of A

Answer 5

If A is any nxn matrix and C(ij) is the cofactor of a(ij). Then the corresponding nxn matrix, where the entries are the cofactors coresponding to the entries of A, is called the matrix of A. The transpose of this matrix is called the adjoint of A.

Question 6

If A is invertible then inv(A) can be expressed as

Answer 6

Inv(A) = 1/det(A)*adj(A)

Question 7

Cramer’s Rule

Answer 7

If Ax=b is a systen of n linear equations in n unknowns such that det(A) != 0, then the system has a unique solution. The solution is

X(1) = det(A(1))/det(A)
X(2) = det(A(2))/det(A)
.
.
.
X(n) = det(A(n))/det(A)

Where A(n) is the matrix obtained by replacing the entries in the n:th column of A by the entries in B.

Question 8

Equivalent statements (3) (6)
A is invertible iff

Answer 8

1) Ax=0 only has the trivial solution
2) The reduced row echelon form of A is I
3) A is expressible as a product of elementary matrices
4) Ax=b is consistent for every nx1 matrix b
5) Ax=b has exactly one solution for every nx1 matrix b
6) det(A) != 0