Matrices are a useful way to represent and manipulate data in order to model situations. The elements in a matrix can represent data, equations or systems and have many real-life applications.

What are matrices?

• A matrix is a?rectangular array?of?elements?(numerical or algebraic) that are arranged in?rows?and?columns
• The?order?of a matrix is defined by the?number?of rows and columns that it has
  • The order of a matrix with ?rows and ?columns is
• A matrix ?can be defined by ?where ?and ?and ?refers to the element in row , column

What type of matrices are there?

• A?column matrix?(or column vector) is a matrix with a?single column,
• A?row matrix?is a matrix with a?single row,
• A?square matrix?is one in which the number of rows is?equal to the number of columns,
• Two matrices are?equal?when they are of the?same order?and their?corresponding elements?are?equal, i.e. for all elements
• A?zero matrix, , is a matrix in which all the elements are o, e.g.
• The identity matrix, , is a?square?matrix in which all elements along the?leading diagonal are 1?and the rest are o, e.g.

What is the transpose of a matrix?

• The?transpose?of matrix?A is denoted as
• The?transpose?matrix is formed by which the rows and columns

Just as with ordinary numbers,?matrices?can be?added?together and?subtracted?from one another, provided that they meet certain conditions.

How is addition and subtraction performed with matrices?

• Two matrices of the?same order?can be added or subtracted
• Only?corresponding elements?of the two matrices are added or subtracted
  • 
• The?resultant?matrix is of the?same order?as the original matrices being added or subtracted

What are the properties of matrix addition and subtraction?

• (commutative)
• (associative)
• 
• 
• 

How do I multiply a matrix by a scalar?

• Multiply?each element?in the matrix by the?scalar?value
  • 
• The?resultant?matrix is of the?same order?as the original matrix
• Multiplication by a?negative?scalar changes the?sign?of each element in the matrix

Can I always multiply a matrix by another matrix?

• Not always - only if the dimensions of the matrices allow it
• If ?has order and?has order then you the matrix ?exists only if
• The order of the matrix?AB?will be
• It is possible for ?to exist but not exist and vice versa
• AB?and?BA?both will exist if they are both?square matrices?of the same order
  • This means the dimensions are the same

How do I multiply a matrix by another matrix?

• To multiply a matrix by another matrix, the?number of columns?in the?first?matrix must be?equal?to the?number of rows?in the?second?matrix
• If the order of the?first matrix isand the order of the second matrix is , then the order of the resultant matrix will be
• The product of two matrices is found by multiplying the corresponding elements in the?row?of the?first?matrix with the corresponding elements in the?column?of the?second?matrix and finding the?sum?to place in the resultant matrix
  • E.g. If
    • then
    • then

How do I square an expression involving matrices?

• If an expression involving matrices is squared then you are multiplying the expression by itself, so write it out in bracket form first, e.g.
  • remember, the regular rules of algebra do not apply here and you cannot expand these brackets, instead, add together the matrices inside the brackets and then multiply the matrices together

What are the properties of matrix multiplication?