2.4: Properties of Matrix Multiplication

A First Course in Linear Algebra · LibreTexts

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    From LibreTexts, under CC BY 4.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    As pointed out above, it is sometimes possible to multiply matrices in one order but not in the other order. However, even if both \(AB\) and \(BA\) are defined, they may not be equal.

    Example \(\PageIndex{1}\): Matrix Multiplication is Not Commutative

    Compare the products \(AB\) and \(BA\), for matrices \(A = \left[ \begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array} \right], B= \left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right]\)

    Solution

    First, notice that \(A\) and \(B\) are both of size \(2 \times 2\). Therefore, both products \(AB\) and \(BA\) are defined. The first product, \(AB\) is

    \[AB = \left[ \begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array} \right] \left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right] = \left[ \begin{array}{rr} 2 & 1 \\ 4 & 3 \end{array} \right] \nonumber\]

    The second product, \(BA\) is

    \[\left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right] \left[ \begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array} \right] = \left[ \begin{array}{rr} 3 & 4 \\ 1 & 2 \end{array} \right] \nonumber\]

    Therefore, \(AB \neq BA\).

    This example illustrates that you cannot assume \(AB=BA\) even when multiplication is defined in both orders. If for some matrices \(A\) and \(B\) it is true that \(AB=BA\), then we say that \(A\) and \(B\) commute. This is one important property of matrix multiplication.

    The following are other important properties of matrix multiplication. Notice that these properties hold only when the size of matrices are such that the products are defined.

    Proposition \(\PageIndex{1}\): Properties of Matrix Multiplication

    The following hold for matrices \(A,B,\) and \(C\) and for scalars \(r\) and \(s\),

    \[ \begin{align} A\left( rB+sC\right) &= r\left( AB\right) +s\left( AC\right) \label{matrixproperties1} \\[4pt] \left( B+C\right) A &=BA+CA \label{matrixproperties2} \\[4pt] A\left( BC\right) &=\left( AB\right) C \label{matrixproperties3} \end{align}\]

    Proof

    First we will prove \(\eqref{matrixproperties1}\). We will use Definition 2.3.1 and prove this statement using the \(ij^{th}\) entries of a matrix. Therefore,

    \[\begin{aligned} \left( A\left( rB+sC\right) \right) _{ij} &=\sum_{k}a_{ik}\left( rB+sC\right) _{kj} \\[4pt] &= \sum_{k}a_{ik}\left( rb_{kj}+sc_{kj}\right) \\[4pt] &=r\sum_{k}a_{ik}b_{kj}+s\sum_{k}a_{ik}c_{kj} \\[4pt] &=r\left( AB\right) _{ij}+s\left( AC\right) _{ij} \\[4pt] &=\left( r\left( AB\right) +s\left( AC\right) \right) _{ij} \end{aligned}\]

    Thus \(A\left( rB+sC\right) =r(AB)+s(AC) \) as claimed.

    The proof of Equation \(\eqref{matrixproperties2}\) follows the same pattern and is left as an exercise.

    Statement Equation \(\eqref{matrixproperties3}\) is the associative law of multiplication. Using Definition 2.3.1,

    \[ \begin{align*}\left( A\left( BC\right) \right) _{ij} &=\sum_{k}a_{ik}\left( BC\right) _{kj} \\[4pt] &=\sum_{k}a_{ik}\sum_{l}b_{kl}c_{lj} \\[4pt] &=\sum_{l}\left( AB\right) _{il}c_{lj}=\left( \left( AB\right) C\right) _{ij}. \end{align*}\]

    This proves \(\eqref{matrixproperties3}\).

    This page titled 2.4: Properties of Matrix Multiplication was last modified on Sun, 27 Feb 2022 23:22:45 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Ken Kuttler (Lyryx) .