Matrix Algebra Operations
In this chapter, we will discuss the elementary operations of addition, subtraction, multiplication, and inversion, a matrix algebra analogue of division of scalar numbers. Subsequently, we will turn our attention to powers of matrices and to operations involving scalar numbers and matrices, scalar numbers and vectors, and vectors and matrices. The operations on matrices differ from similar operations of scalar algebra in several respects. The matrix algebra operations, in general, are not commutative and attention must be paid to whether the matrices are conformable with respect to the intended operation. Also, it must be noted whether the matrix operation pertains to matrix elements or to matrices.
Addition of Matrix Elements
To add elements of matrices A and B and store the result as a matrix C,
elements of matrix A is are added to their corresponding elements in matrix B and stored as elements of matrix C. Obviously, all three matrices must be the same size. Notice that the plus sign in the above equation is enclosed in parentheses to indicate addition of matrix elements, as contrasted with addition of matrices, to be discussed in section to follow. An example of addition of matrix elements is schematized below
and illustrated as
Subtraction of Matrix Elements
The operation of subtracting matrix elements can be schematized as
and illustrated as
Addition of Matrices
To add two matrices A and B and store the results in matrix C
the number of rows in matrix A must equal the number of columns in matrix B and the number of rows in matrix B must equal the number of columns in matrix A, in another words, the matrices must be conformable to matrix addition. The schematic representation of matrix addition is
illustrated as
Subtraction of Matrices
To subtract two matrices A and B and store the results in matrix C
again, the matrices must be conformable. The schematic representation of matrix subtraction is shown below
illustrated as
The operation of addition and subtraction can be also performed on vectors. Consider a column vector X = [a b c]. It's major addition
can be symbolically represented as
and illustrated as
The minor subtraction of the column vector X
can be symbolically represented as
and illustrated as
Since the matrix and the vector must be conformable to addition, a matrix can be added only to a row vector and only a column vector can be added to a matrix. The same is true of subtraction of matrices and vectors. In the special case of addition or subtraction of a zero vector and a matrix, both operations of subtraction and addition are identical. The addition (or subtraction) of a row zero vector and a matrix
can be symbolically represented as
and illustrated as
results in a column sum of matrix elements. The addition (or subtraction) of a matrix and a column zero vector
can be symbolically represented as
and illustrated as
results in row sums of matrix elements.
Multiplication of Matrix Elements
To multiply elements of two matrices A and B and store the results in matrix C
each element of matrix A is multiplied with its corresponding element in matrix B and the result is stored in matrix C. To perform this operation, both matrices have to be either of the same size, or one matrix has to be a scalar matrix. The schematic representation of multiplication of matrix elements is
illustrated by a special case of multiplication of a matrix by a scalar number as
in the special case of multiplication of a matrix by a constant it is customary to omit the (.) sign.
Multiplication of Matrices
The matrix multiplication is analogous to matrix addition, and, as the matrix addition, this operation is not commutative, unless both matrices are symmetric. Also, to premultiply matrix B by matrix A
the matrices must be conformable to matrix multiplication. The product matrix C will have the number of rows of the first matrix and the number of columns of the second matrix. The schematic representation of matrix multiplication is
illustrated as
The postmultiplicatin of matrix B by matrix A
using the same numerical example, is illustrated as
Notice that the product of any matrix with its own transpose is always square. The multiplication of a symmetric matrix and its transpose is commutative. The transpose of a matrix product is equal to the product of the transposes of the original matrices multiplied in reverse order. For example, (AB)'=B'A'.
Premultiplication of a matrix A by a diagonal matrix T
is equivalent to scalar multiplication of each row element in the matrix by the same row element of the diagonal matrix. In schematic representation
The following example illustrates rearrangement of the columns of a matrix
Division of Matrix Elements
To divide elements of two matrices A and B and store the results in matrix C
corresponding elements of A and B matrices form a fraction, stored in C. All three matrices must be of the same size, or the divisor must be a scalar number. The division sign is enclosed in parentheses to indicate division of matrix elements. An example of division of matrix elements is schematized below
and illustrated, using as an example a division of a matrix by a scalar number
Matrix Inversion
The analog of matrix division is multiplication by a reciprocal. The reciprocal of a matrix is called its inverse. Only square matrices can be inverted, however, some square matrices, called singular matrices, do not have inverses. The inverse of a matrix is denoted by a -1 superscript. To invert a matrix, first, we must determine that the matrix is invertible by computing its determinant. If the determinant is not a zero, the matrix is invertible. This is analogous to the restriction on the division of scalar numbers that cannot be divided by a zero. Consider a special case of a two by two matrix A
with determinant equal to ad-bc.
For the special case of the two-by-two matrices, the inverse 
of the above matrix can be calculated by
switching elements in the principal diagonal,
changing signs of the off-diagonal elements,
and by dividing all elements by the determinant.
Consider a matrix A
First, we must determine whether this matrix is invertible. The determinant is computed as (1)(4)-(2)(3) and equals -2. This determinant does not equal zero. It means that the matrix is invertible.
To invert a matrix, switch elements in the principal diagonal,
change signs of the off-diagonal elements,
and divide all elements by the determinant
Thus, the matrix inverse equals
Analogous to scalar multiplication of a number by its reciprocal
an identity matrix I equals
this analogy also suggests the nature of matrix singularity. In arithmetic, the only number without a reciprocal is 0. In matrix algebra, the singular matrices, i.e., matrices with a zero determinant, do not have an inverse. A numerical illustration of the above equation is
The discussed procedures for matrix inversion apply only to two-by-two matrices. The inverse of a matrix larger than a two-by-two matrix is a complicated operation which is best done with the help of a computer.
Powers of Matrix Elements
The squares of the matrix elements are written as
In schematic representation
A numerical example is
A
matrix to a fractional power 
provides
for obtaining roots of matrix elements and a matrix to a fractional negative
power 
for obtaining reciprocals of roots of matrix
elements.
Powers of Matrices
The square of a matrix is a multiplication of a matrix by itself
