How to calculate determinant of 3x3 matrix

rewrite first two columns then calculate product of \+\+\ and subtract /-/-/

What happens to determinant if we change two rows or columns

the determinant change its sign

What is property of determinant 1, in traspose matrix 2, multiplying every element by k

1, in traspose matrix is the same as original matrix det(A)=det(A´) 2, multiplying every element by k it multiplying by k^n where n is dimension of matrix det(kA)= k^n det(A)

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) = Π diagonal elements

How to calculate the inverse matrix

for each element b(i,j) b(i,j)= 1/ det(A) * (-1)^(i+j) * det(Aij) where (Aij) is the determinant of matrix without current row and column

What is linear combination

linear combination = scaled vectors added together (0) (1)* 3 & (1) (0)*2 = (2) (3)

What is linear dependence of vectors

when c1v1+c2v2+..ckvk=0 if at least ine vector can be build as combination or multiplication of others vectors (only for non-trivial so c≠0)

What is rank of the 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-sqaure 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 sqaure submatrix whose derminant is not zero >> its size gives the rank

How to solve linear equation with matrix notation

A.. matrix with n x k x ... unknown coefficient vector with k x 1 b... result vector with n x 1

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AMM #2

Question	Answer
How to calculate determinant of 2x2 matrix	det(M)= ad - bc
How to calculate determinant of 3x3 matrix	rewrite first two columns then calculate product of \+\+\ and subtract /-/-/
What happens to determinant if we change two rows or columns	the determinant change its sign
What is property of determinant 1, in traspose matrix 2, multiplying every element by k	1, in traspose matrix is the same as original matrix det(A)=det(A´) 2, multiplying every element by k it multiplying by k^n where n is dimension of matrix det(kA)= k^n det(A)
What happend to matrix determinant if we add/subtract other rows/column 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) = Π diagonal elements
How to calculate the inverse matrix	for each element b(i,j) b(i,j)= 1/ det(A) * (-1)^(i+j) * det(Aij) where (Aij) is the determinant of matrix without current row and column
What is linear combination	linear combination = scaled vectors added together (0) (1)* 3 & (1) (0)*2 = (2) (3)
What is linear dependence of vectors	when c1v1+c2v2+..ckvk=0 if at least ine vector can be build as combination or multiplication of others vectors (only for non-trivial so c≠0)
What is rank of the 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-sqaure 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 sqaure submatrix whose derminant is not zero >> its size gives the rank
How to solve linear equation with matrix notation	A.. matrix with n x k x ... unknown coefficient vector with k x 1 b... result vector with n x 1

Created by: semester1.IBWL

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