Matrices

Question 1

What do i and j represent in matrices?

Answer 1

i is the row and j is the column.

Question 2

What is the identity matrix? What is a null matrix?

Answer 2

The identity matrix is a matrix which has 0’s everywhere except diagonally from the top left to bottom right, which are 1’s. The null matrix is just a matrix full of 0’s.

Question 3

What is the transpose of a matrix?

Answer 3

Where aji = aij, the matrix is symmetric.

Question 4

What order matrix do you get when you multiply two matrix together?

Answer 4

A matrix with the number of rows equal to that of matrix A and a number of columns equal to that of matrix B.

Question 5

How do you find the elements of the product of two matrices A and B?

Answer 5

cij = sum from k=1 to number of columns in A or number of rows in B of :aik*bkj

Question 6

What do you get if you multiply a matrix A by the identity matrix I?

Answer 6

You get the original matrix, A.

Question 7

What do you get if you multiply a matrix by its inverse?

Answer 7

The identity matrix I.

Question 8

What is the equation for the inverse of a matrix including minors?

Answer 8

A^-1 = 1/|A| * C(transpose)

where C is the cofactor matrix whose elements are multiplied by (-1)^(i+j)

Question 9

How do you find the inverse of a 2 by 2 matrix?

Answer 9

• Swap the position of the two leading diagonal elements
• Negate the two off-diagonal elements
• Multiply all by the reciprocal of the determinant of tha matrix

Question 10

How do you find the inverse of a 3 by 3 matrix?

Answer 10

Find all the minors - this is the cofactor matrix.

Question 11

How do you solve simultateous equations using the inverse matrix?

Answer 11

Put the equations into a matrix, with one unknown column and one column for the answers. The answers to the simultaneous equation are the inverse of the matrix multiplied by the answers of the equation. (e.g. x+y = 4 - 4 is the answer here)

Question 12

How do you solve simultaneous equations using determinants?

Answer 12

• Solve the equations to find x and y
• The answers you get should look similar to determinants
• Convert to determinants and rearrange so you have x and y on the top and the determinants as the denominators

Question 13

How do you solve simultaneous equations using Gaussian elimination?

Answer 13

Use row operations to make a triangle of 0’s in the bottom left corner, starting from the top and working round the corner

Question 14

What does it mean if, during Gaussian elimination, you get a zero in the bottom right corner (whole bottom row are zeros?

Answer 14

There is no solution to this set of equations.

Question 15

How do you solve simultaneous equations using row reduced echelon form?

Answer 15

Use row operations until you have the identity matrix. Find zeros in same order, but after bottom left triangle, start from top middle zero and then work round the corner.

Question 16

What does it mean if a set of vectors, e.g. a, b and c are linearly independent?

Answer 16

Linearly independent if αa + βb + γc = 0 is only satisfied if α = β = γ = 0

Question 17

What is the procedure to work out if a set of vectors is linearly independent?

Answer 17

Put α, β and γ in front of each matrix and make it equal to zero, and find α, β and γ. Then form a matrix with the vectors as columns and obtain row-reduced echeclon form. If there are all zeros in final column, is linearly independant.

Question 18

What are the conditions required for a set of vectors to form a linear vector space?

Answer 18

• The set is closed under commutative and associative addition (a+b = b+a and (a+b)+c = a+(b+c)
• The set is closed under multiplication by a scalar that gives a new vector
• There exists a null vector such that a=0 = a
• Multiplication by unity leaves all vectors unchanged
• All vectors have a corresponding negative vector such that: a+(-a) = 0 and -1 x a = -a

Question 19

What is the span of a set of vectors?

Answer 19

The infinity set of vectors that may be written as linear combinations of the original set of vectors.

Question 20

What is the image of a vector A?

Answer 20

The span of the column vectors of A.

Question 21

What is the kernel of a vector A?

Answer 21

The solution of the square matrix multiplied by α, β and γ, all equaling zero (vector with α, β and γ as x,y and z)

Question 22

What is the rank of a matrix A?

Answer 22

The dimension of the image of A (number of linearly independent rows or linearly independent columns). Is also the number of non-zero rows in row reduced echelonform. ALso the size of the largest square submatrix with a non-zero determinant.

Question 23

What is nullity of matrix A?

Answer 23

The dimension of the kernel of A.

Question 24

What is the rank-nullity theorem?

Answer 24

For a m by n matrix, rank(A) + nullity(A) = n

Question 25

How can you tell if there are solutions from the rank?

Answer 25

If the rank is equal to the number of columns of A, then there are unique solutions. If the rank of A is different to the rank of the cofactor matrix then there are no solutions.

Question 26

What are linear transformations?

Answer 26

Where a matrix is multiplied by another matrix A to transform it into another vector space.

Question 27

How do you find the eigenvalues of a matrix?

Answer 27

Find the magnitude of the matrix - lambda x the identity matrix. Then solve the equation.

Question 28

How do you find the eigenvectors?

Answer 28

Multiply the found matrix from (A-lambda X I) by e11 and e21, using the different eigenvalues.

Question 29

How do you normalise eigenvectors?

Answer 29

Multiply the found eigenvector by beta which is equal to the inverse of the square root of the sum of the individual elements squared.

Question 30

How can you find eigenvectors using the vectors/cross product?

Answer 30

Find (A-lambda X I) and take the vector product of the first two rows.

Question 31

How do you find the inverse of a matrix using the identity matrix?

Answer 31

Perform row operations until matrix A is in row-reduced echelon form, but do the same row operations to the identity matrix. This will convert I into the inverse of A.

Question 32

What is matrix diagonalisation?

Answer 32

You want to find the non-singular matrix P such that: D = P^-1 X A X P, where D is a diagonal matrix.

Question 33

How do you find P?

Answer 33

P is constructed by arranging the eigenvectors of A as the columns. The diagonal elements of D then correspond to the eigenvalues of A.

Question 34

What do you get if you put a Diagonal matrix to the power of n?

Answer 34

All the diagonal elements are put to the power of n too.