Matrices
Year 11 (IGCSE) 🔢 Number Matrix addition, multiplication, determinant and inverse.
🔢 Matrix Basics
A matrix is a rectangular array of numbers. The order is stated as rows × columns.
Example: $\mathbf{A} = \begin{pmatrix} 2 & -1 \ 3 & 4 \end{pmatrix}$ is a $2 \times 2$ matrix (2 rows, 2 columns).
Add or subtract matrices by combining corresponding elements — only possible when both matrices have the same order.
⚡ Identity Matrix
$$\mathbf{I} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \quad \text{(satisfies } \mathbf{AI} = \mathbf{IA} = \mathbf{A}\text{)}$$✖️ Matrix Multiplication
To multiply matrices, the number of columns in the first must equal the number of rows in the second. Compute each element as a row-times-column dot product.
⚡ 2×2 Multiplication
$$\begin{pmatrix}a&b\c&d\end{pmatrix}\begin{pmatrix}e&f\g&h\end{pmatrix} = \begin{pmatrix}ae+bg & af+bh\ce+dg & cf+dh\end{pmatrix}$$Example: $\begin{pmatrix}2&1\3&0\end{pmatrix}\begin{pmatrix}4\-1\end{pmatrix} = \begin{pmatrix}8-1\12+0\end{pmatrix} = \begin{pmatrix}7\12\end{pmatrix}$
🔄 The Inverse and Determinant
The inverse matrix $\mathbf{A}^{-1}$ satisfies $\mathbf{A}\mathbf{A}^{-1} = \mathbf{I}$. It exists only when the determinant is non-zero.
⚡ 2×2 Inverse
$$\mathbf{A} = \begin{pmatrix}a&b\c&d\end{pmatrix} \Rightarrow \mathbf{A}^{-1} = \frac{1}{ad-bc}\begin{pmatrix}d&-b\-c&a\end{pmatrix}$$The determinant is $\det(\mathbf{A}) = ad - bc$.Example: $\mathbf{A} = \begin{pmatrix}3&1\5&2\end{pmatrix}$. $\det = 6-5=1$. $\mathbf{A}^{-1} = \begin{pmatrix}2&-1\-5&3\end{pmatrix}$
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