MATH 54 Lecture Notes 2

1 Matrix Arithmetic

1.1 Multiplying Matrices and Vectors

Let A be an m×n matrix with entries a_ij for 1≤i≤m and 1≤j≤n, and let
b be an n element column vector with entries b_j for 1≤j≤n. Then c = Ab
is an m element column vector. If we call the entries of c by c_i for 1≤i≤m,
then

Some examples:

If the number of columns of the matrix is not equal to the number of entries
of the vector, the product is not defined .

1.2 Multiplying Matrices

Let A be an m×n matrix with entries a_ij and B be an n×q matrix with entires
b_jk. Then C = AB is an m × q matrix. If we denote its entries by c_ij , where
1≤i≤m and 1≤j≤q, then

Another way of putting this is to write B as a matrix of column vectors:

where each b_k is an n element vector, the j-th entry of which is b_jk. Then

where each Ab_k, being a matrix times a column vector, can be computed using
(1).

Some examples:

Note that matrix multiplication is not commutative.
If the number of columns of the first matrix is not equal to the number of
rows of the second matrix, then the product is not defined.
We can also multiply matrices by scalars (that is, by elements of R or C).
To do this, simply multiply each entry of the matrix by the given scalar.

1.3 Adding Matrices

Let A and B both be m × n matrices, with entries a_ij and b_ij , respectively.
Then C = A + B is also an m × n matrix, with entries

Some examples:

If the matrices do not have the same dimensions (that is, the same number
of rows and the same number of columns), then the sum is not defined .

1.4 Some Special Matrices

1.4.1 The Zero Matrix

The matrix 0 is a matrix of any dimension with every entry equal to 0. (The
dimension should be chosen to make things work.) Multiplying 0 by any matrix
or vector gives 0, except possibly with different dimensions . This matrix is also
the additive identity.

1.4.2 The Identity Matrix

For any n, the n × n identity matrix, written I_n, is the matrix with every
diagonal entry equal to 1 and every other entry equal to 0. Any matrix or
vector multiplied by I_n, on either the left or the right, will be the same matrix
or vector (as long as n is chosen appropriately).

2 Elementary Matrices and Matrix Inverses

2.1 Definition of the Inverse

If A is an n × n matrix, then A⁻¹ is an n × n matrix such that

If A has an inverse, then we say that A is invertible. It only makes sense to
talk about the inverses of square matrices .
Some facts about inverse matrices:

• If a matrix has the correct size and is an inverse of A on the left (or on
the right), then that matrix is A⁻¹.
• Matrix inverses are unique, when they exist.
• If A and B are invertible n × n matrices, then so is AB, and

Inverses are not as well-behaved with respect to addition. In particular, you
can add two non -invertible matrices and get an invertible matrix, or add two
invertible matrices and get a non-invertible matrix.

2.2 Elementary Matrices

Each of the three types of elementary row operations can be performed by
multiplication on the left by a square matrix. These matrices are invertible.
For example,

multiplies the third row by −5. The matrix

inter changes rows 1 and 2, and the matrix

adds −3 times the first row to the second row.

2.3 Computing Inverses

Let A be some n × n matrix. Suppose E₁,E₂, . . . ,E_r are elementary matrices
such that E₁E₂ · · ·E_rA = I. Then A⁻¹ = E₁E₂ · · ·E_r. Therefore, whatever
row ope rations turn A into I, those same row operations turn I into A⁻¹.