Matrix involved qs

Question 1

Finding the inverse of a matric via determinant formula

Answer 1

A^-1= 1/detA *(adjA)

Question 2

Finding the inverse of a matrix via determinant process

Answer 2

Calculate determinant and adjA, then combine, knowing the rule for determinant for 3*3

Question 3

How to find adjA

Answer 3

Make chessboard of + and -s, like for determinant then crossword determinat for each value

Question 4

Calculating the determinant

Answer 4

We use +-+ for the top and cross off with values and corners then calculate

Question 5

Gaussian elimination

Answer 5

Process by manipulating rows, with edges = to ans then we make it look like the identity matrix (I) to calculate x, y and z

Question 6

Gaussian elimination for finding inverse

Answer 6

Rather than being equal to 0s we make rhs equal to Identity matrix, we then manipulate left to look like the identity matrix whilst also altering the identity matrix the same way (manipulating the rows of I)

Question 7

LU decomposition

Answer 7

Process by which we split our matrix into an upper and lower matrix, our upper matrix will have common top line and 0’s below diagonal, lower will have 0’s in the upper and 1’s across diagonal

Question 8

LU decomposition method

Answer 8

Draw identity matrix multiplying original matrix.
Have top line constant then move down making the lower triangle of 0s for the upper via row operations.
If we do a row operation taking away 3R3, we must add 3R3 to the identity as they are being multiplied.

Question 9

Eigenvalues and eigenvectors equation

Answer 9

Ax = hx
Where x is a vector and h is a scalr value, and A is our original matrix.
h is our eigenvalue and x is our eigenvector.

Question 10

Finding eigenvalues process

Answer 10

Rewrite equation and move to LHS so RHS = 0.
Multiply by Identity matrix and take x out as a common factor.
Determinant of bracket (A-hI) = 0
Calculate det(A-hI), keeping in our values of h as unknow, this will generate a polynomial when set to 0. Solve the polynomial for our eigenvalues.

Question 11

Finding eigenvectors process

Answer 11

Input each eigenvalue into our bracket equation, use gaussian elimination to solve for x1,x2,x3 etc. at 0, these values will be our eigenvector (we will use 1 as our value for x1 if it is not 0)