Edexcel IAL - Further Pure Mathematics 1- 5.1 Matrix Addition and Subtraction- Study notes - New syllabus
Edexcel IAL – Further Pure Mathematics 1- 5.1 Matrix Addition and Subtraction -Study notes- New syllabus
Edexcel IAL – Further Pure Mathematics 1- 5.1 Matrix Addition and Subtraction -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 5.1 Matrix Addition and Subtraction
Addition and Subtraction of Matrices
A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used to store and manipulate data in a structured form.
Two matrices can only be added or subtracted if they have the same order, that is, the same number of rows and columns.
Definition
Let
\( A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}, \quad B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} \)
Then:
\( A + B = \begin{pmatrix} a_{11}+b_{11} & a_{12}+b_{12} \\ a_{21}+b_{21} & a_{22}+b_{22} \end{pmatrix} \)
\( A – B = \begin{pmatrix} a_{11}-b_{11} & a_{12}-b_{12} \\ a_{21}-b_{21} & a_{22}-b_{22} \end{pmatrix} \)
This means we add or subtract corresponding entries.
Properties
- Matrix addition is commutative: \( A + B = B + A \).
- Matrix addition is associative: \( (A + B) + C = A + (B + C) \).
- There is a zero matrix \( O \) such that \( A + O = A \).
- For any matrix \( A \), \( A – A = O \).
Example
\( A = \begin{pmatrix} 2 & 5 \\ 1 & -3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & -1 \\ 3 & 2 \end{pmatrix} \)
Find \( A + B \).
▶️ Answer / Explanation
\( A + B = \begin{pmatrix} 2+4 & 5+(-1) \\ 1+3 & -3+2 \end{pmatrix} = \begin{pmatrix} 6 & 4 \\ 4 & -1 \end{pmatrix} \)
Example
\( A = \begin{pmatrix} 3 & -2 & 1 \\ 0 & 4 & 5 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 6 & -3 \\ 2 & -1 & 4 \end{pmatrix} \)
Find \( A – B \).
▶️ Answer / Explanation
\( A – B = \begin{pmatrix} 3-1 & -2-6 & 1-(-3) \\ 0-2 & 4-(-1) & 5-4 \end{pmatrix} = \begin{pmatrix} 2 & -8 & 4 \\ -2 & 5 & 1 \end{pmatrix} \)
Example
\( A = \begin{pmatrix} x & 2 \\ 3 & y \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 1 \\ -2 & 4 \end{pmatrix} \)
If \( A + B = \begin{pmatrix} 8 & 3 \\ 1 & 10 \end{pmatrix} \), find \( x \) and \( y \).
▶️ Answer / Explanation
Compare corresponding entries:
\( x + 5 = 8 \Rightarrow x = 3 \)
\( 2 + 1 = 3 \)
\( 3 + (-2) = 1 \)
\( y + 4 = 10 \Rightarrow y = 6 \)