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a rectangular array of elements (or entries) set out by rows and columns
vertical linear array of numbers
a hortizontal array of numbers side by side
zero matrix
matrix containing zeros
diagonal matrix
has zeros in all positions other than the leading diagonal which contains numbers other than zero.
What are the objects in a set called?
plural of matrix
(m X n) matrix
A matrix with m rows and n columns
B + B + B
Subtraction of matrix
subtracting corresponding elements
adding and subtracting of matrices
only on matrices of the same order
multiplication by a scalar (number)
each element is multiplied by the scalar
equality of matrices
matrices are equal if they are of the same order and all corresponding elements are equal
identity matrix
diagonal matrix with ones in the leading diagonal and zeros in the other positions.
square matrix
the same number of elements in the rows and columns.
A X B exists and order is m X p
A = m X n and B = n X p, then A X B
A is a square matrix and I is the identity matrix
inverse matrix
AX = B
X = A^-1X B
XA = B
X = BA^-1
the numbers in a matrix
dimensions, example
2 by 3
what does the 2 represent in 2x3
what does the 3 represent in 2x3
row matrix
a matrix has only 1 row
column matrix
a matrix has only 1 column
how are two matrices equal?
if their dimensions are the same and the entries in corresponding positions are equal
the dimensions have to be the same
what must occur in order for matrices to be able to be added and subtracted?
what is the communitive property?
axb is the same as bxa
are matrices communitive?
what is the associative property?
a(bc) is the same as (ab)c
are matrices associative?
det A or |A|
how is a determinant denoted by?
keep the area positive
when finding the area of the triangle, what do you always do?
5+0=5 or 5x1=5
what is the identity property?
5+ (-5)= 0
what is the inverse property?
multiply by 1 0
in a 2x2 matrix, how do you get the multiplicative identity?
multiply by 1 0 0
in a 3x3 matrix, how do you get the multiplicative identity?
addition and substraction
add or subtract corresponding entries- cant add/ subtract with unequal dimensions
equal matrices
-dimensions are the same
properties of matrix addition
commutative property of matrix addition
matrix multiplication
-can be multiplied if the # of A columns = # of B rows
does not hold true
commutative property of matrix multiplication
Additive Identity
The additive identity is 0. a + 0 = 0 + a = a
additive inverse
the opposite of a number
Multiplicative Identity
A * 1 = A
multiplicative inverse
reciprocal of a number
2x2 (cross multiply and subtract)
additive identity matrices
multiply the matrix by a matrix with all zeros
additive inverse matrices
multiply the matrix by a matrix by the same matrix except all entries are negated
multiplicative identity matrices
only for square matrices- (nxn) with a diagonal of 1s from left to right
multiplicative inverse matrices
1/det [d -b -c a]
solving matrix equa
[A] X = [B] (multiply A(-1) times B = X)
solving systems with matrices
8x + 5y = 2
a movement on the cartesian plane
[1 0, 0 -1]
reflective about the x-axis transformation
[-1 0, 0 1]
reflective about the y-axis transformation
a transformation that changes the size of an object, but not the shape
90* rotation
switch the x and y values and negate the x values [0 -1, 1 0]
180* rotation
negate all values -1[ ]
270* rotation
switch the x and y values and negate the y values [0 1, -1 0]
same [1 0, 0 1]