Adding Matrices & Multiplying by a Scalar (Edexcel IGCSE Maths)

Basic operations with matrices

Spec point — 5C

How do I add or subtract two matrices?

• To add or subtract two matrices, they need to be of the same order (i.e. the same number of rows and columns)

• If they are of the same order, then

  • to add two matrices, just add the corresponding elements

    • $( \begin{gathered} 2 - 3 \\ 5 1 \end{gathered} ) + ( \begin{gathered} - 1 & 7 \\ - 4 & 6 \end{gathered} ) = ( \begin{gathered} 2 + ( - 1 ) & - 3 + 7 \\ 5 + ( - 4 ) & 1 + 6 \end{gathered} ) = ( \begin{gathered} 1 4 \\ 1 7 \end{gathered} )$

  • to subtract two matrices just subtract the corresponding elements

    • $( \begin{gathered} 2 - 3 \\ 5 1 \end{gathered} ) - ( \begin{gathered} - 1 & 7 \\ - 4 & 6 \end{gathered} ) = ( \begin{gathered} 2 - ( - 1 ) & - 3 - 7 \\ 5 - ( - 4 ) & 1 - 6 \end{gathered} ) = ( \begin{gathered} 3 - 10 \\ 9 - 5 \end{gathered} )$

How do I multiply a matrix by a scalar?

• To multiply any matrix by a scalar (a number), multiply each element by that scalar

  • If $A = ( \begin{gathered} 5 2 \\ 0 4 \end{gathered} )$ then $2 A = 2 ( \begin{gathered} 5 2 \\ 0 4 \end{gathered} ) = ( \begin{gathered} 2 \times 5 & 2 \times 2 \\ 2 \times 0 & 2 \times 4 \end{gathered} ) = ( \begin{gathered} 10 4 \\ 0 8 \end{gathered} )$

• Lower case letters often refer to scalar multiples

  • $kA$ is the matrix $A$multiplied by the scalar $k$

Worked Example $A = ( \begin{gathered} - 2 & 2 \\ - 2 & - 5 \end{gathered} ) B = ( \begin{gathered} - 4 & 1 \\ 3 - 4 \end{gathered} )$

(a) Calculate $A+B$

Answer:

Add the corresponding elements of the two matrices

$\begin{gathered} A + B & = ( \begin{gathered} - 2 & 2 \\ - 2 & - 5 \end{gathered} ) + ( \begin{gathered} - 4 & 1 \\ 3 - 4 \end{gathered} ) \\ = ( \begin{gathered} - 2 + ( - 4 ) & 2 + 1 \\ - 2 + 3 & - 5 + ( - 4 ) \end{gathered} ) \\ = ( \begin{gathered} - 6 & 3 \\ 1 - 9 \end{gathered} ) \end{gathered}$

Final answer: $\begin{gathered} A + B & = ( \begin{gathered} - 6 & 3 \\ 1 - 9 \end{gathered} ) \end{gathered}$

Given that $2 A - n B = ( \begin{gathered} 8 1 \\ - 13 & 2 \end{gathered} )$

(b) find the value of $n$

Answer:

Multiply all elements of $A$ by 2 to find $2A$

$\begin{gathered} 2 A & = 2 ( \begin{gathered} - 2 & 2 \\ - 2 & - 5 \end{gathered} ) \\ = ( \begin{gathered} 2 \times ( - 2 ) & 2 \times 2 \\ 2 \times ( - 2 ) & 2 \times ( - 5 ) \end{gathered} ) \\ = ( \begin{gathered} - 4 & 4 \\ - 4 & - 10 \end{gathered} ) \end{gathered}$

Multiply all elements of $B$ by $n$ to find $nB$

$\begin{gathered} n B & = n ( \begin{gathered} - 4 & 1 \\ 3 - 4 \end{gathered} ) \\ = ( \begin{gathered} - 4 n & n \\ 3 n & - 4 n \end{gathered} ) \end{gathered}$

Subtract the corresponding elements to find $2A-nB$

$\begin{gathered} 2 A - n B & = ( \begin{gathered} - 4 & 4 \\ - 4 & - 10 \end{gathered} ) - ( \begin{gathered} - 4 n & n \\ 3 n & - 4 n \end{gathered} ) \\ = ( \begin{gathered} - 4 - ( - 4 n ) & 4 - n \\ - 4 - 3 n & - 10 - ( - 4 n ) \end{gathered} ) \\ = ( \begin{gathered} 4 n - 4 & 4 - n \\ - 3 n - 4 & 4 n - 10 \end{gathered} ) \end{gathered}$

Substitute that into the equation from the question

$\begin{gathered} ( \begin{gathered} 4 n - 4 & 4 - n \\ - 3 n - 4 & 4 n - 10 \end{gathered} ) & = ( \begin{gathered} 8 1 \\ - 13 & 2 \end{gathered} ) \end{gathered}$

For those two matrices to be equal, the correct value of $n$ must make ALL the corresponding elements equal in the two matrices

• So you can choose any one pair to solve to find $n$

$\begin{gathered} 4 n - 4 & = 8 \\ 4 n & = 12 \\ n = 3 \end{gathered}$

It is worth checking the other three pairs of elements with that value of $n$, to make sure you haven't made a mistake

$4 - 3 = 1 ✔

• 3 ( 3 ) - 4 = - 13 ✔ 4 ( 3 ) - 10 = 2 ✔$

$n=3$