Skip to main content
Engineering LibreTexts

2.6: Matrix Operations

  • Page ID
    44467
  • A matrix is a special type of array with well-defined algebraic laws for equality, addition, subtraction, and multiplication. Matrices are defined as the set of coefficients of a system of linear algebraic equations or as the coefficients of a linear coordinate transformation. A linear algebraic system of three equations is given by

    \[{3 x + 4 y - 2 z=-2} \\ {8 x - 2 y - 7 z=5} \label{38} \\ {5 x - 3 y + 4 z=3} \]

    In matrix form, we express this system of equations according to

    \[\begin{bmatrix} {3} & { 4} & {-2} \\ {8} & {-2} & {-7} \\ {5} & {-3} & { 4} \end{bmatrix} \begin{bmatrix} {x} \\ {y} \\ {z} \end{bmatrix} = \begin{bmatrix} {-2} \\ { 5} \\ { 3} \end{bmatrix} \label{39}\]

    The rule for multiplication implied here is that the first row of the \(3\times 3\) matrix multiplies the \(3\times 1\) unknown column matrix to obtain the first element of the \(3\times 1\) known column matrix. In terms of compact notation, we express Equation \ref{39} in the form13

    \[\mathbf{Au} = \mathbf{b} \label{40}\]

    Here \(\mathbf{ A}\) represents the \(3\times 3\) matrix in Equation \ref{39}

    \[\mathbf{ A}=\begin{bmatrix} {3} & {4} & {-2} \\ {8} & {-2} & {-7} \\ {5} & {-3} & {4} \end{bmatrix} \label{41}\]

    while \(\mathbf{u}\) and \(\mathbf{b}\) represent the two \(3\times 1\) column matrices according to

    \[\mathbf{ u}=\begin{bmatrix} {x} \\ {y} \\ {z} \end{bmatrix} , \qquad \mathbf{ b}=\begin{bmatrix} {-2} \\ { 5} \\ { 3} \end{bmatrix} \label{42}\]

    The \(3\times 1\) matrices are sometimes called column vectors; however, the word vector should be reserved for a quantity that has magnitude and a direction, such as a force, a velocity, or an acceleration. In this text, we will use the phrase column matrix for a \(n\times 1\) matrix and row matrix for a \(1\times n\) matrix. Examples of a \(1\times 4\) row matrix and a \(4\times 1\) column matrix are given by

    \[\mathbf{ b}=\begin{bmatrix} {3} & {1} & {5} & {6} \end{bmatrix} , \qquad \mathbf{ c}=\begin{bmatrix} { 6} \\ {-2} \\ { 0} \\ { 4} \end{bmatrix} \label{43}\]

    In a matrix, the numbers are ordered in a rectangular grid of rows and columns, and we indicate the size of an array by the number of rows and columns. The following is a \(4\times 4\) matrix denoted by \(\mathbf{ A}\):

    \[\mathbf{ A}=\begin{bmatrix} {3} & { 1} & { 5} & { 6} \\ {4} & {-3} & { 6} & {-2} \\ {8} & { 3} & {-2} & { 0} \\ {1} & { 5} & { 8} & { 4} \end{bmatrix} \label{44}\]

    Matrices have a well-defined algebra that we will explore in more detail in subsequent chapters. At this point we will introduce only the operations of scalar multiplication, addition and subtraction. Scalar multiplication of a matrix consists of multiplying each element of the matrix by a scalar, thus if \(c\) is any real number, the scalar multiple of Equation \ref{44} is given by

    \[c \mathbf{ A}=\begin{bmatrix} {3c} & { 1c} & { 5c} & { 6c} \\ {4c} & {-3c} & { 6c} & {-2c} \\ {8c} & { 3c} & {-2c} & {0} \\ {1c} & { 5c} & { 8c} & { 4c} \end{bmatrix} \label{45}\]

    Matrices have the same size when they have the same number of rows and columns. For example, the two matrices \(\mathbf{A}\) and \(\mathbf{B}\)

    \[\mathbf{ A}=\begin{bmatrix} {a_{11} } & {a_{12} } & {......} & {a_{1n} } \\ {a_{21} } & {a_{22} } & {......} & {a_{2n} } \\ {....} & {....} & {......} & {....} \\ {a_{m1} } & {a_{m2} } & {......} & {a_{mn} } \end{bmatrix} , \qquad \mathbf{ B}=\begin{bmatrix} {b_{11} } & {b_{12} } & {......} & {b_{1n} } \\ {b_{21} } & {b_{22} } & {......} & {b_{2n} } \\ {....} & {....} & {......} & {....} \\ {b_{m1} } & {b_{m2} } & {......} & {b_{mn} } \end{bmatrix} \label{46}\]

    have the same size and the sum of \(\mathbf{A}\) and \(\mathbf{B}\) is created by adding the corresponding elements to obtain

    \[\mathbf{ A} + \mathbf{ B}=\begin{bmatrix} {a_{11} +b_{11} } & {a_{12} +b_{12} } & {......} & {a_{1n} +b_{1n} } \\ {a_{21} +b_{21} } & {a_{22} +b_{22} } & {......} & {a_{2n} +b_{2n} } \\ {....} & {....} & {......} & {....} \\ {a_{m1} +b_{m1} } & {a_{m2} +b_{m2} } & {......} & {a_{mn} +b_{mn} } \end{bmatrix} \label{47}\]

    It should be apparent that the sum of two matrices of different size is not defined, and that subtraction is carried out in the obvious manner indicated by

    \[\mathbf{ A} - \mathbf{ B}=\begin{bmatrix} {a_{11} -b_{11} } & {a_{12} -b_{12} } & {......} & {a_{1n} -b_{1n} } \\ {a_{21} -b_{21} } & {a_{22} -b_{22} } & {......} & {a_{2n} -b_{2n} } \\ {....} & {....} & {......} & {....} \\ {a_{m1} -b_{m1} } & {a_{m2} -b_{m2} } & {......} & {a_{mn} -b_{mn} } \end{bmatrix} \label{48}\]

    When working with large matrices, addition and subtraction is best carried out using computers and the appropriate software.

    • Was this article helpful?