Intro to Matrices

1. It's important to remember that the multiplication of matrices is not commutative, i.e. A(inv).A <> A.A(inv).

1'. The product of matrices is associative as long as the order of the matrices does not change i.e. (AB)C = A(BC)

1". The product of matrices is distributive over addition i.e. A(B+C) = AB + AC notice that (A+B)C = AC + BC

2. To check if the matrix is inversible,

a. ensure it is a square matrix,

a. make sure that it's determinent is not null i.e. det(A) <> 0 which means that matrix has full rank

i. no zero column or row,

ii. no column (row) that is equal to another column (row)

iii. no column (row) is linear combination of the remaining comlumns (rows) of the matrix

3. Ex.p2, 1. XA=B, XA.inv(A) = B.inv(A), XI=B.inv(A), X=B.inv(A)

4. Ex.p2, 2. to check if B is inversible, we calc. det(B), to calc. det(B), we:

a. augment B with the 1st 2 cols.

b. draw a 3 elements downward diagonal rays starting from the leading entry (1st. el. in 1st. row and 1st. col. b11)

c. draw 2 parallel 3 els. downward diagonals

d. start with the last el. in the 1st. col. and draw a 3 els. upward diagonal

e. draw 2 parallel 3 els. upward diagonals

f. mullt. the els. of each diagonal and write them at the bottom of the downward diagonal or the top of the upward diagonal

g. add up the 3 bottom products and substract from the sum the sum of the 3 top products

5. The inverse of the product of 2 matrices is the same as the product of the 2 inverted matrices, w.r.t. the order of the matrices i.e. inv(AB) = inv(A)*inv(B)

5'. The inverse of the inversed matrix is the original matrix itself i.e. inv(inv(A)) = A

5". The inverse of the product of 2 matrices is the product of the inverse of each matrix in reverse order i.e. inv(AB) = inv(B) * inv(A)

5"'.If matrix is inveversible any puissance of the matrix is inversible & it doesn't matter if we calc. the inverse 1st or the puissance i.e. inv(A cube)=cube(inv(A))

6. The inverse of the transposed matrix is the same as the transpose of the inverted matrix i.e. inv(transp(A)) = transp(inv(A))

6'. The transpose of the transposed matrix is the original matrix itself i.e. transp(transp(A)) = A

6". The transpose of the sum of 2 matrices is the sum of the transpose of each matrix i.e. transp(A+B) = transp(A) + transp(B)

7. Matrices are used to solve linear equations A.X = U, where A is the matrix of coeff. of the inconnu X and U is the matrix of constant value on the right hand side of the equation

8. The linear system represented by AX = U has a

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