matrices

Question 1

what is the order of a matrix with a rows and b columns ?

Answer 1

order a,b

Ma,b

Question 2

what are the numbers in the matrix called ?

Answer 2

elements

Question 3

the element in matrix A or row i and column j is noted as ?

Answer 3

Aij

Question 4

in a matrix ( 1 2 )
( 3 4 )

what is the leading diagonal ?

Answer 4

1 ,4

Question 5

what is the upper triangular in a matrix ?

Answer 5

all elements where i > j

Question 6

what is the lower triangular in a matrix ?

Answer 6

all elements where i < j

Question 7

what is the transpose of a matrix ?

Answer 7

A = A^t

the i,j th element becomes the j,i th element

Amn becomes Anm

Question 8

how can you tell when a matrix is symmetric ?

Answer 8

A = A^T

Question 9

what is a

a. row vector ?
b. column vector ?

Answer 9

a. a vector with only 1 row

b. a vector with only 1 column

Question 10

how do you add/subtract matrices A and B ?

Answer 10

you add/subtract each corresponding element ?

like : A11 +/- B11

Question 11

how do you multiply a matrix by a constant ?

Answer 11

you multiply each element by the constant

e.g. A11 x K and A12 x K

Question 12

how do you multiply two matrices ?

Answer 12

you can only multiply matrices where the first matrix has the same number of columns and the second matrix has rows

you multiply the first matrix rose by the first column of the second matrix and add those elements together

every time you start a new row of the first matrix , start a new row in the answer matrix

every time you start a new column of the second matrix , start a new column in the answer column

Question 13

what is this property of a matrix : commutative law ?

Answer 13

AB /= BA

Question 14

what is this property of a matrix : associative law ?

Answer 14

A(BC) = (AB)C

Question 15

what is this property of a matrix : distributive law ?

Answer 15

• scalar multiple
  K(AB) = (KA)B = A(KB) - over addition
  A(B+C) = AB + AC

Question 16

what is this property of a matrix : multiplication by identity matrix ?

Answer 16

IA = A = AI

Question 17

what is this property of a matrix : transpose of sum ?

Answer 17

(A+B)^T = A^T + B^T

Question 18

what is this property of a matrix : transpose of a product ?

Answer 18

(AB)^T = A^T.B^T

Question 19

what is the determinant of a 2x2 matrix ?

Answer 19

the product of the leading diagonal - the product of the second diagonal

if |A| = 0 the A is a singular matrix and it does not have an inverse

Question 20

what is the inverse of a 2x2 matrix ?

Answer 20

1/the determinant . the original matrix (with the leading diagonal elements swapped and the elements on the second diagonal changed signs)

Question 21

how do you solve simultaneous equations using matrices ?e

Answer 21

put the equations in the form AX=B

then times both sides by A^-1

and that solves it

X = A^-1 . B

Question 22

how of you calculate the determinant of a 3x3 matrix ( any row ) ?

Answer 22

choose the row and
times each element by the determinant of the minor of the element.

then if the order adds to an even number don’t change the sign but if its odd, change the sign.

then add the elements

Question 23

what is the minor of an element ?

Answer 23

the determinant of the elements left when you eliminate all the elements in the same row and column of your chosen element.

Question 24

if the two rows or columns in a matrix are identical what is the determinant ?

Answer 24

zero

Question 25

how do you calculate the determinant of a matrix with the rule of sarrus ?

Answer 25

write out the matrix
next to it write the two first columns again
circle the diagonals of the top three elements and add them all together
circle the diagonals of the bottom three elements and add them all together
then subtract the second number from the first number
this is the determinant

Question 26

what is the co factor of an element ?

Answer 26

it is the determinant of the minor with the associated sign ( the sign of the original element, except if the order adds to an odd number , change the sign )

if the matrix is A the cofactor matrix is C

Question 27

how do you solve simultaneous equations with three unknown variables, using matrices ?

Answer 27

again put them in the form AX=B
find the determinant of A = |A|
find the matrix of cofactors C
transpose c to give C^t ( also known as the adjoint )
find the inverse of A by A^-1 = 1/|A| . C^T
multiply B . A^-1 to give you X

Question 28

what is cramer’s rule ?

Answer 28

you take a A

and to find a variable you put |A|/|A| BUT replace the column of the variable you are trying to find (x=1,y=2,z=3) with the B matrix and solve to find the answer