Section 1: Reading a Matrix

Given a matrix A that has m rows and n columns, then the entry in the ith row and jth column of A is a_ij and is called the (i, j) – entry of A:

	Column j
Row i		= A

Section 2: Matrix addition and scalar multiplication

Two matrices can only be added if they are of equal size, meaning both matrix A and matrix B have m rows and n columns.

Example:

A + C = Error because the 3rd column of A cannot be matched to anyting in C

Addition and Scalar Multiplication Theorems:

If A, B, and C be matrices of the same size, and r and s are scalars.

1. A + B = B + A
2. B. (A + B) + C = A + (B + C)
3. A + 0 = A
4. r(A + B) = rA + rB
5. (r + s)A = rA + sA
6. r(sA) = (rs)A

Section 3: Matrix multiplication

Definition: If A is an m x n matrix, and if B is an n x p matrix with columns b₁, . . . , b_p, then the product AB is the m x p matrix whose columns are Ab₁, . . . , Ab_p, that is,

AB = A[b₁ b₂ . . . b_p] = [Ab₁ Ab₂ . . . Ab_p]

Move mouse over the numbers to see an example

Row-Column Rule for Computing AB: If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding entries from row i of A and column j of B. If (AB)_ij denotes the (i,j)-entry in AB, and if A is an m x n matrix, then:

Move mouse over the answer matrix to see an example

Matrix Multiplication Theorems:

If A is a matrix with m rows and n columns, and matrices B and C have sizes that do not produce errors for the following examples, then:

1. A(BC)= (AB)C
2. A(B + C) = AB + Ac
3. (B + C)A = BA + CA
4. r(AB) = (rA)B = A(rB)
5. A = A = AI_n

IMPORTANT: For a matrix A to be multiplied with matrix B, the number of columns in A MUST be equal to the number of rows in matrix B.

This means that:

1. AB may not always be equal to BA (move mouse over numbers to see reason):



2. Cancelation laws are not true for matrix multiplication: if AB = AC, then it is not always true that B = C
3. If the product of AB is a zero matrix, it is not always true that A = 0 or B = 0

Section 4: Matrix Powers

If A is a square matrix with dimensions n x n, then A^k is the product of k comies of A. If k=0, then A⁰ is the identity matrix of A.

Section 5: Transpose of Matrix

If A is an m x n matrix, the transpose of A is an n x m matrix denoted by A^T where the columns are formed by the rows of A.

Example:

Transpose of Matrix Theorems

1. (A^T)^T = A
2. (A + B)^T = A^T + A^T
3. For any scalar r, (rA)^T = rA^T
4. (AB)^T = B^TA^T

Section 6: Partitioned Matrices  

One of the key things in linear algebra is the ability to split up a bigger matrix into smaller subsections, which is also known as partitioning a matrix.  For an instance, we have a matrix A:

Can be partitioned into its subsections based on the user’s discretion. Here is a partitioned Matrix:

These subsections make up the Matrix:

Section 7: Addition and Scalar Multiplication of Partitioned Matrices

Addition works the same way with partition matrices.

+

As long as the matrices are partitioned in the same manner and have the same size, the laws of addition for matrices still hold true.

==>

Scalar Multiplication on a partition matrix works the same way as multiplication of a scalar on a regular matrix as well.

Section 8: Multiplication of Partitioned Matrices

The purpose of partition up a matrix for multiplication is to split up a bigger matrix into smaller chunks, which makes it easier to compute. For an instance, take this example:

or

becomes:

and

The Multiplication still works the same way, just within the partitions themselves:

==>

The key is that when you substitute back in the numbers, matrices are being substituted back in and not just a single number.

Substituting the equations back in gets you:

And

Which leads to:

If A is m x n and B is n x p, then

It is from these components that you multiply out and get other matrices then you add up all the individual bits to get the full answer

(C) 2004 Jin Tao