Multiplying Matrices (Edexcel International A Level (IAL) Further Maths: Further Pure 1): Revision Note

Multiplying Matrices

How do I multiply a 2x2 matrix by a 2x1 matrix?

• Multiply the corresponding elements in the row of the first matrix with the corresponding elements in the column of the second matrix, writing their sum in the answer matrix

• The answer will be a 2 × 1 matrix

  • $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}ax+by\\ cx+dy\end{array}\right)$

• $\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)\left(\begin{array}{c}10\\ 20\end{array}\right)=\left(\begin{array}{c}1\times 10+2\times 20\\ 3\times 10+4\times 20\end{array}\right)=\left(\begin{array}{c}10+40\\ 30+80\end{array}\right)=\left(\begin{array}{c}50\\ 110\end{array}\right)$

How do I multiply a 2x2 matrix by another 2x2 matrix?

• Multiply the corresponding elements in the row of the first matrix with the corresponding elements in the column of the second matrix, writing their sum in the answer matrix 

• The answer will be a 2 × 2 matrix

  • $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)=\left(\begin{array}{cc}aA+bC& aB+bD\\ cA+dC& cB+dD\end{array}\right)$

• $\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)\left(\begin{array}{cc}5& 10\\ 20& 25\end{array}\right)=\left(\begin{array}{cc}1\times 5+2\times 20& 1\times 10+2\times 25\\ 3\times 5+4\times 20& 3\times 10+4\times 25\end{array}\right)=\left(\begin{array}{cc}5+40& 10+50\\ 15+80& 30+100\end{array}\right)=\left(\begin{array}{cc}45& 60\\ 95& 130\end{array}\right)$

• This process becomes more natural the more times you do it!

How do I square a matrix?

• Do not square each individual element inside the matrix

• Write out a matrix multiplication

  • If $\mathbf{P}=\left(\begin{array}{cc}2& 4\\ 1& -3\end{array}\right)$ then ${\mathbf{P}}^2=\mathbf{P}\times \mathbf{P}=\left(\begin{array}{cc}2& 4\\ 1& -3\end{array}\right)\left(\begin{array}{cc}2& 4\\ 1& -3\end{array}\right)=\left(\begin{array}{cc}2\times 2+4\times 1& 2\times 4+4\times \left(-3\right)\\ 1\times 2+\left(-3\right)\times 1& 1\times 4+\left(-3\right)\times \left(-3\right)\end{array}\right)=\left(\begin{array}{cc}8& -4\\ -1& 13\end{array}\right)$

• It is possible to have negative elements in a squared matrix

How do I multiply matrices of any dimensions?

• 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

  • For example, the first matrix is $m\times n$ and the second matrix is $n\times p$

  • The order of the resultant matrix will be $m\times p$

• Multiply corresponding elements in the row of the first matrix with the corresponding elements in the column of the second matrix

  • Place their sums in the resultant matrix

    • For example, if $\mathbf{A}=\left(\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right)$, $\mathbf{B}=\left(\begin{array}{cc}g& h\\ i& j\\ k& l\end{array}\right)$

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

    • whereas $\mathbf{BA}=\left(\begin{array}{ccc}(ga+hd)& (gb+he)& (gc+hf)\\ (ia+jd)& (ib+je)& (ic+jf)\\ (ka+ld)& (kb+le)& (kc+lf)\end{array}\right)$

• It is possible for AB to exist but BA not to exist

  • For square matrices of the same order, AB and BA will both exist

What does commutative mean?

• Commutative means "swapping the order doesn't change the result"

  • 5 × 4 = 4 × 5 and 3 + 2 = 2 + 3

    • Multiplication and addition of numbers are commutative

  • 4 ÷ 2 ≠ 2 ÷ 4 and 5 - 3 ≠ 3 - 5

    • Division and subtraction of numbers are not commutative

• However, matrix multiplication is not commutative

  • In general AB ≠ BA

• For example, $\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)\left(\begin{array}{cc}0& 1\\ 5& 1\end{array}\right)=\left(\begin{array}{cc}10& 3\\ 20& 7\end{array}\right)$ but $\left(\begin{array}{cc}0& 1\\ 5& 1\end{array}\right)\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)=\left(\begin{array}{cc}3& 4\\ 8& 14\end{array}\right)$

What does associative mean?

• Associative means "it doesn't matter which order you group operations into"

  • To do 5 + 4 + 3, either (5 + 4) + 3 or 5 + (4 + 3) works

  • To do 8 × 9 × 10, either (8 × 9) × 10 or 8 × (9 × 10) works 

    • Multiplication and addition of numbers are associative

  • (8 ÷ 4) ÷ 2 ≠ 8 ÷ (4 ÷ 2) and (5 - 4) - 3 ≠ 5 - (4 - 3)

    • Division and subtraction of numbers are not associative

• Matrix multiplication is associative

  • (AB)C ≡ A(BC)

• To multiply three matrices together, it is fine to either:

  • start by multiplying the first two together,

  • or start by multiplying the second two together

    • Just don't switch the order

    • A(BC) is not the same as (BC)A

The Identity Matrix

What is the Identity Matrix?

• The identity matrix, $\mathbf{I}$, is a square matrix with:

  • Ones along the leading diagonal (from top-left to bottom-right)

  • and zeros everywhere else

    • The 2 × 2 identity matrix is $\mathbf{I}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

    • The 3 × 3 identity matrix is $\mathbf{I}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

  • The notation ${\mathbf{I}}_n$ can be used to specify the $n\times n$ identity matrix

• Multiplying any square matrix by the same-sized identity matrix leaves it unchanged
  

  • Both $\mathbf{AI}=\mathbf{A}$ and $\mathbf{IA}=\mathbf{A}$

  • $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ and $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$

    • This result can be proved by multiplying together the matrices