An Introduction to Matrices

The Definition of a Matrix and Matrix Terminology.

An \(m\times n\) matrix is a rectangle of numbers (or functions) arranged in \(m\) rows and \(n\) columns in square brackets. Entries are indexed by two subscripts, the first being the row, the second the column:

For brevity, we typically represent this matrix as \([a_{ij}]\) (rather than writing all the entries).

Two matrices are equal if they have the same size and like entries are equal.

Some special matrices:

• If \(m=n\) then \(A\) is said to be a square matrix.
• A matrix whose entries are all zeros is called a zero matrix.
• A square matrix of size \(n\) with \(1\)'s down the diagonal and zeros elsewhere (i.e., \(a_{ij}=1\) if \(i=j\) and \(a_{ij}=0\) otherwise) is called the identity matrix and is denoted by \(I_{n}\text{.}\)

Operations on Matrices.

Suppose that \(A=[a_{ij}]\) and \(B=[b_{ij}]\) are two matrices and \(\alpha\) is a number. We define:

• Matrix Addition: When \(A\) and \(B\) are both \(m\times n\) matrices, we define \(A+B=[a_{ij}+b_{ij}]\)
• Matrix Subtraction: When \(A\) and \(B\) are both \(m\times n\) matrices, we define \(A-B=[a_{ij}-b_{ij}]\)
• Scalar Multiplication: For any matrix \(A\text{,}\) we define \(\alpha A=[\alpha a_{ij}]\text{.}\)
• Matrix Multiplication: when \(A\) is \(m\times n\) and \(B\) is \(n\times p\) we define \(AB=[c_{ij}]\) where \(c_{ij}=a_{i1}b_{1j} +a_{i2}b_{2j} +\dots +a_{in}b_{nj}\) for \(i=1,\dots m\) and \(j=1\dots ,p\text{.}\) Note: the \(ij\)-th entry in the matrix \(AB\) is equal to the dot product of the \(i\)th row of \(A\) and the \(j\)th column of \(B\text{.}\)

• Determinants: When \(A\) is a square matrix, we define its determinant iteratively as follows:

  • For a \(2\times 2\) matrix \(A=\left[ \begin{array}{cc} a_{11} \amp a_{12} \\ a_{21} \amp a_{22} \end{array} \right]\) we define the determinant of \(A\) written \(\det(A)\) or \(|A|\) to be the number
    \begin{equation*} \det(A) =a_{11}a_{22}-a_{21}a_{12}\text{.} \end{equation*}
  • For a \(3\times 3\) matrix \(A=\left[ \begin{array}{ccc} a_{11} \amp a_{12} \amp a_{13} \\ a_{21} \amp a_{22} \amp a_{23} \\ a_{31} \amp a_{32} \amp a_{33} \\ \end{array} \right]\) we define
    \begin{equation*} \det(A) =a_{11} \det\left( \left[ \begin{array}{cc} a_{22} \amp a_{23} \\ a_{32} \amp a_{33} \end{array} \right] \right) -a_{12} \det \left( \left[ \begin{array}{cc} a_{21} \amp a_{23} \\ a_{31} \amp a_{33} \end{array} \right] \right) +a_{13} \det \left( \left[ \begin{array}{cc} a_{21} \amp a_{22} \\ a_{31} \amp a_{32} \end{array} \right] \right) \end{equation*}
    i.e. a \(3\times 3\) determinant is calculated by using \(2\times 2\) determinants.
  • Higher dimensional determinants are defined in an analogous way.

Applications of Matrices: Linear Transformations.

For an \(n\times n\) matrix \(A\text{,}\) we define a function \(F_{A}\) whose domain consists of vectors (or points) in \({\mathbb R}^n\) (considered as \(n\times 1\) matrices) and whose range is also vectors (or points) in \({\mathbb R}^n\) (considered as \(n\times 1\) matrices) by \(F_{A}({\bf v}) =A\bf v.\) We call \(F_{A}\) a linear transformation of \({\mathbb R}^n\text{.}\)

• The absolute value of a determinant tells us how much the area of an object is scaled after the linear transformation is applied.
• The sign of the determinant tells us whether the transformation preserves orientation (essentially, preserving orientation means that the linear transformation has not caused a reflection).