matrices Flashcards
what is the order of a matrix with a rows and b columns ?
order a,b
Ma,b
what are the numbers in the matrix called ?
elements
the element in matrix A or row i and column j is noted as ?
Aij
in a matrix ( 1 2 )
( 3 4 )
what is the leading diagonal ?
1 ,4
what is the upper triangular in a matrix ?
all elements where i > j
what is the lower triangular in a matrix ?
all elements where i < j
what is the transpose of a matrix ?
A = A^t
the i,j th element becomes the j,i th element
Amn becomes Anm
how can you tell when a matrix is symmetric ?
A = A^T
what is a
a. row vector ?
b. column vector ?
a. a vector with only 1 row
b. a vector with only 1 column
how do you add/subtract matrices A and B ?
you add/subtract each corresponding element ?
like : A11 +/- B11
how do you multiply a matrix by a constant ?
you multiply each element by the constant
e.g. A11 x K and A12 x K
how do you multiply two matrices ?
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
what is this property of a matrix : commutative law ?
AB /= BA
what is this property of a matrix : associative law ?
A(BC) = (AB)C
what is this property of a matrix : distributive law ?
- scalar multiple
K(AB) = (KA)B = A(KB) - over addition
A(B+C) = AB + AC
what is this property of a matrix : multiplication by identity matrix ?
IA = A = AI
what is this property of a matrix : transpose of sum ?
(A+B)^T = A^T + B^T
what is this property of a matrix : transpose of a product ?
(AB)^T = A^T.B^T
what is the determinant of a 2x2 matrix ?
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
what is the inverse of a 2x2 matrix ?
1/the determinant . the original matrix (with the leading diagonal elements swapped and the elements on the second diagonal changed signs)
how do you solve simultaneous equations using matrices ?e
put the equations in the form AX=B
then times both sides by A^-1
and that solves it
X = A^-1 . B
how of you calculate the determinant of a 3x3 matrix ( any row ) ?
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
what is the minor of an element ?
the determinant of the elements left when you eliminate all the elements in the same row and column of your chosen element.
if the two rows or columns in a matrix are identical what is the determinant ?
zero
how do you calculate the determinant of a matrix with the rule of sarrus ?
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
what is the co factor of an element ?
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
how do you solve simultaneous equations with three unknown variables, using matrices ?
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
what is cramer’s rule ?
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
