MATRIX DEFINITIONS

Def'n: a matrix is a rectangular array of numbers known as entries. Each entry has an
address to indicate its position in the array -- row first, then column, so a _{i j} is the entry
in the ith row and jth column of matrix A.

is a 2 % 3 matrix. a _{2 1} = –7

We use caps (upper case letters) to name matrices and corresponding lower case letters to name the entries. If the matrix name is B, then we denote its entries as b _{i j} with i = row #, j = col #.

Def'n: A matrix A with n rows and n columns is called a square matrix of order n and the entries a₁₁, a₂₂, ..... a_nn are on the main diagonal of A.

is a 3 % 3 matrix. a _{1 1} = 1, a _{2 2} = 5, and a _{3 3} = 4.

Def'n: A matrix in which all entries off the main diagonal are 0 is called a diagonal matrix.

Def'n: Two matrices are said to be equal if they have the same size and their
corresponding entries are equal.

Def'n: If A is an m % n matrix, the transpose of A denoted ( A^t ), is the n % m matrix
formed by turning each row of A into a column with respect to position.

Def'n: a zero matrix is one in which all entries = 0.

** BE CAREFUL: zero matrices don't behave like the real number 0.

in R -- the Real numbers, if ab = ac, and a 0, then b = c, and if ab = 0, then a = 0 or b = 0.

THIS IS NOT NECESSARILY TRUE FOR MATRICES!!

PROPERTIES OF THE ZERO MATRIX assuming the matrix sizes conform
a) A + 0 = 0 + A = A	b) A – A = 0	c) 0 – A = –A	d) A × 0 = 0, 0 × A = 0

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MATRIX OPERATIONS

Adding: If matrices A and B are the same size, then (A + B) is the matrix we get by adding together the corresponding entries in A and B. (A + B) _{i j} = (a _{i j} + b _{i j} ).

Matrices of different sizes cannot be added!!

Scalar Multiplication: If A is a matrix and c is a scalar or constant, the product cA
is the matrix we get by multiplying every entry of A by c.

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Assuming the sizes of the matrices conform, these are the properties of matrix arithmetic:

a) A + B = B + A	commutative property for addition
b) A + (B + C) = (A + B) + C	associative property for addition
c) A(BC) = (AB)C	associative property for multiplication
d) A(B + C) = AB + AC	distributive property on the left for multiplication
e) (B + C)A = BA + CA	distributive property on the right for multiplication
f) a(B + C) = aB + aC	distributive property on the left for scalar multiplication
g) (a + b)C = aC + bC	distributive property on the right for scalar multiplication
h) a(bC) = (ab)C	associative property for scalar multiplication
i) a(BC) = (aB)C = B(aC)	associative property for scalar/ matrix multiplication.

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Matrix Multiplication: If A is m % r and B is r % n, the product AB is an m % n matrix
with ab _{i j} , the ijth entry of AB, equal to the summation of the products we get when
the r entries in the ith row of A multiply the r entries in the jth row of B.

Note: The number of columns in the matrix on the left must equal the number of rows in
matrix on the the right. Since, with matrices, AB is not neccesarily equal to BA, position matters when multiplying matrices. In some cases, AB exists, but BA does not.

Let's do an example to illustrate the process.

Here, A is 2 % 3 , and B is 3 % 2 so both products AB and BA exist.
When we multiply B on the LEFT by A, we get a 2 % 2 matrix, AB.
To get ab _{1 1}: the 3 entries in row 1 of A ( a _{1 1}, a _{1 2}, a _{1 3} ,) multiply the 3 entries in
column 1 of B, then they're added to get: 1(7) + 4(0) – 2(5) = –3.

Similarly, to get ab _{1 2}: the 3 entries in row 1 of A ( a _{1 1}, a _{1 2}, a _{1 3} ,) multiply the 3 entries in
column 2 of B, then they're added to get: 1(–3) + 4(–2) – 2(–1) = – 9.

To get ab _{2 1} , the entry in the 2nd row and 1st column of AB, we multiply the 3 entries
in row 2 of A ( a _{2 1}, a _{2 2}, a _{2 3} ,) by the 3 entries in column 1 of B. We add them to get 40.

The final entry ab _{2 2} , is the sum of the products of the 3 entries in row 2 of A and
column 2 of B.

Since A is 2 % 3 , and B is 3 % 2, we can find BA, a 3 % 3 matrix like this:

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MATRIX MULTIPLICATION INSTRUCTIONS

If matrices A and B are such that product AB exists, then
entry ab _{i j} , in the i-th row and j-th column of product AB,
is the sum of the products found when the entries in the i-th row of A
are multiplied by the entries in the j-th column of B.

Examples: let's multiply A (3 % 2) with B (2 % 2) to get AB (3 % 2).

step 1-1: row-1 × column-1 = (1 × 3) + (2 × – 2) = – 1 -- the entry in ab ₁₁ .

step 1-2: row-1 × column-2 = (1 × – 4) + (2 × 1) = – 2 -- the entry in ab ₁₂ .

step 2-1: row-2 × column-1 = (– 1 × 3) + ( – 3 × – 2) = 3 -- the entry in ab ₂₁ .

step 2-2: row-2 × column-2 = (– 1 × – 4) + ( – 3 × 1) = 1 -- the entry in ab ₂₂ .

step 3-1: row-3 × column-1 = (0 × 3) + ( – 4 × – 2) = 8 -- the entry in ab ₃₁ .

step 3-2: row-3 × column-2 = (0 × – 4) + ( – 4 × 1) = – 4 -- the entry in ab ₃₂ .

Now we'll multiply a 1 % 3 (a row vector) by a 3 % 1 (a column vector) to get a 1 % 1 or scalar.

This value is known as the dot product of 2 vectors. It is used in 3-dimensional geometry to indicate orthogonal (a fancy shmancy Westmount word for perpendicular) vectors and determine the class (acute, right or obtuse) of the angle between 2 vectors. This is where the dot symbol for multiplication comes from.

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Practice

1)

a) find (2A + 3D)

b) find (–5B + 3C)

c) find A t .

d) (2A + 3C)

2) Using the matrices in question #1,

a) state whether these products exist.		b) if yes, say what size the product is.
AB	AC	CD	BC

3)

a) find AB

b) find BA

c) find AC

d) (2BA + 3C)

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Solutions

1)

a) (2A + 3D) =	b) (–5B + 3C) =
c) find A t =	d) (2A + 3C) doesn't exist; sizes are wrong.

2) Using the matrices in question #1,

a) state whether these products exist.		b) if yes, say what size the product is.
AB exists; 3 % 3	AC exists; 3 % 3	CD exists; 2 % 2	BC doesn't exist.

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3)

a)

b)

c)

d)

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