AMM#2

How to calculate determinant of 2×2 matrix

det(M) = ad - bc

How to calculate determinant of 3×3 matrix

rewrite first two columns

then calculate product of \+\+\+& /-/-/-

What happens to determinant if we change two rows or columns

the determinant change its sign

What is property of determinant

1, in transpose matrix

2,multiplying every element by k

1, in transpose matrix it is the same as in original matrix

det(A) = det(A´)

2,multiplying every element by k, then we need to sqaure k by nr. of dimension

det(kA) = k^n * det(A)

What happend to matrix determinant if we add/subtrack other rows/columns within a matrix

it stays unchanged

What is determinant

1,det(I)

2,det(AB)

3,det(A^-1)

4,det(A+B)

5,det(triangular / diagonal)

1,det(I) = 1

2,det(AB) = det(A) * det(B)

3,det(A^-1) = 1/det(A)

4,det(A+B) ≠ det(A) + det(B)

5,det(triangular / diagonal) = ∏ diagonal elements

How to calculate the inverse matrix

for each element b(i,j)

b(i,j) = 1/det(A) * (-1)^(i+j) * det|Aj.i|

What is linear combination

linear combination= scaled vectors added together

(0)

(1) * 3

&

(1)

(2) * 2

=

(2)

(3)

What is linear dependence of vectors

when c1v1+c2v2+ ck+vk = 0

if at least one vector can be build as combination or multiplication of other vectors (only for non-trivial so c≠0)

What is rank of a matrix (3)

1, if matrix is divided into rows or columns how many of them are maximaly independent

2, nr. of ind. row= nr. of ind. col.

3, if non-singular rk(A) = n

Can rank be calculated for non-square matrices

yes, not as determinant it can be calculated also for rectangular matrices

What are 3 properties of matrix rank

1, exchanging rows or columns does not change the rank

2,rk(A) = rk(A´)

3,rk(A) = rk (A´A)

How to determine the rank of the matrix (3)

1, with Eigenvalues

2,with Gaussian algorithm

3, finding the largest square submatrix whose determinant is not zero » its size gives the rank

What are 3 ways to solve linear equation

1, Gaussian Elimination

2, Inverse Matrix

3, Cramers Rule

How to calcualte linear equations with Guassian Eliminations

we transform into [A| b ] and solve with element in every line

How to calculate linear equation with Inverse Matrix

we calculate A^-1 and the multiply it with solution vector

How to calculate linear equation with Cramers rule

1, we calculate det(M)

2, for each column we replace it with solution vector and then

xn = det(M with column n) / det(M)

How to calculate rank of matrix with Gaussian elimination

we try to calculate the triangle like shape (similar to linear system) and count linear with zeros only