Operations with Matrices (DP IB Applications & Interpretation (AI): HL): Revision Note

Matrix addition & subtraction

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

  • $\mathit{A}\pm \mathit{B}=\left(a_{ij}\right)\pm \left(b_{ij}\right)=(a_{ij}\pm b_{ij})$

    • e.g. $\left(\begin{array}{cc}1& 0\\ 2& -1\end{array}\right)+\left(\begin{array}{cc}2& -1\\ 3& 1\end{array}\right)=\left(\begin{array}{cc}3& -1\\ 5& 0\end{array}\right)$

• The resultant matrix is of the same order as the original matrices being added or subtracted

What are the properties of matrix addition?

• Addition is commutative

  • $\mathit{A}+\mathit{B}=\mathit{B}+\mathit{A}$

• Addition is associative

  • $\mathit{A}+(\mathit{B}+\mathit{C})=(\mathit{A}+\mathit{B})+\mathit{C}$

• The zero matrix is the additive identity

  • $\mathit{A}+\mathit{O}=\mathit{A}$ for all matrices $\mathit{A}$

• Subtraction is the inverse of addition

  • $\mathit{O}-\mathit{A}=-\mathit{A}$

  • $\mathit{A}-\mathit{B}=\mathit{A}+(-\mathit{B})$

Matrix multiplication

How do I multiply a matrix by a scalar?

• Multiply each element in the matrix by the scalar value

  • $k\mathit{A}=\left(k{a}_{ij}\right)$

    • e.g. $3\left(\begin{array}{cc}2& 0\\ 1& 2\\ -1& 3\end{array}\right)=\left(\begin{array}{cc}6& 0\\ 3& 6\\ -3& 9\end{array}\right)$

• 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

How do I multiply a matrix by another matrix?

• You can only multiply two matrices if their orders are compatible

  • $\mathit{A}\times \mathit{B}$ only exists if the number of columns in $\mathit{A}$ is equal to the number of rows in $\mathit{B}$

• If the order of $\mathit{A}$ is $m\times n$ and the order of $\mathit{B}$ is $n\times p$ then the order of $\mathit{A}\times \mathit{B}$ is $m\times p$

• To find the element of $\mathit{A}\times \mathit{B}$in the row $i$ and the column $j$

  • multiply each element in the row $i$of matrix $\mathit{A}$with the corresponding element in the column $j$ of matrix $\mathit{B}$

  • find the sum of the products

    • e.g. If $\mathit{A}=\left[\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right]$, $\mathit{B}=\left[\begin{array}{cc}g& h\\ i& j\\ k& l\end{array}\right]$

      • then $\mathit{A}\mathit{B}=\left[\begin{array}{cc}(ag+bi+ck)& (ah+bj+cl)\\ (dg+ei+fk)& (dh+ej+fl)\end{array}\right]$ 

      • then  $\mathit{B}\mathit{A}=\left[\begin{array}{ccc}(ga+hd)& (gb+he)& (gc+hf)\\ (ia+jd)& (ib+je)& (ic+jf)\\ (ka+ld)& (kb+le)& (kc+lf)\end{array}\right]$

What are the properties of matrix multiplication?

• Multiplication is not commutative

  • In general $\mathit{A}\mathit{B}\ne \mathit{B}\mathit{A}$

  • Order matters

• Multiplication is associative

  • $\mathit{A}\left(\mathit{B}\mathit{C}\right)=\left(\mathit{A}\mathit{B}\right)\mathit{C}$

• Multiplication is distributive with addition

  • $\mathit{A}(\mathit{B}+\mathit{C})=\mathit{A}\mathit{B}+\mathit{A}\mathit{C}$

  • $(\mathit{A}+\mathit{B})\mathit{C}=\mathit{A}\mathit{C}+\mathit{B}\mathit{C}$

• The identity matrix is the multiplicative identity

  • $\mathit{A}\mathit{I}=\mathit{I}\mathit{A}=\mathit{A}$

• Any product involving a zero matrix results in a zero matrix

  • $\mathit{A}\mathit{O}=\mathit{O}\mathit{A}=\mathit{O}$

  • The converse is not true

    • i.e. if $\mathit{A}\times \mathit{B}=\mathbf{0}$ then it is possible that both $\mathit{A}$ and $\mathit{B}$ are non-zero

• Powers of square matrices are repeated multiplication

  • ${\mathit{A}}^2=\mathit{A}\mathit{A}$

  • ${\mathit{A}}^3=\mathit{A}\mathit{A}\mathit{A}$

  • etc