# Matrix algebra

> Edexcel International A-Level Further Mathematics · IAL FM FP1 2018
> Source: https://www.owlsprep.com/study/edexcel-ial-further-math-fp1-matrix-algebra/

This guide covers all core 2x2 matrix algebra content required for Edexcel IAL Further Maths FP1, including arithmetic operations, determinants, singular/non-singular classification, and inverses with exam-aligned worked examples.

**Prerequisites:** Basic algebraic rearrangement skills

## Learning objectives

- Perform matrix addition, subtraction and scalar multiplication correctly for 2×2 matrices
- Compute products of conformable 2×2 matrices, and explain why matrix multiplication is non-commutative
- Evaluate 2×2 determinants, and classify matrices as singular or non-singular
- Calculate the inverse of non-singular 2×2 matrices using the standard formula
- Apply the reverse order rule for the inverse of a product: $(AB)^{-1} = B^{-1}A^{-1}$

## Matrix Addition, Subtraction & Scalar Multiplication

Matrices are rectangular arrays of numbers, with dimensions given as rows × columns. For FP1, all matrices you will work with are 2×2. Two matrices can be added or subtracted only if they have the same dimensions: you add/subtract corresponding elements individually.

**Scalar multiplication** — Multiplying every element of a matrix by a constant (scalar) value.

**Worked example:** Given $A = \begin{pmatrix} 2 & 1 \\ -3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & -2 \\ 0 & 7 \end{pmatrix}$, calculate $2A - B$.

1. Compute 2A by multiplying each element of A by 2:

   $$2A = \begin{pmatrix} 2\times 2 & 2\times 1 \\ 2\times (-3) & 2\times 4 \end{pmatrix} = \begin{pmatrix} 4 & 2 \\ -6 & 8 \end{pmatrix}$$
2. Subtract corresponding elements of B from 2A:

   $$2A - B = \begin{pmatrix} 4-5 & 2 - (-2) \\ -6 - 0 & 8 -7 \end{pmatrix} = \begin{pmatrix} -1 & 4 \\ -6 & 1 \end{pmatrix}$$

> **Exam tip:** Always check dimensions first before attempting addition or subtraction; if matrices are not the same size, the operation is undefined.

*Calculator:* allowed

## Matrix Multiplication

Matrix multiplication follows a dot product rule, and is only defined when the number of columns in the first matrix equals the number of rows in the second matrix (conformable). For 2×2 matrices $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $B = \begin{pmatrix} e & f \\ g & h \end{pmatrix}$, the product $AB = \begin{pmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{pmatrix}$.

> **warning**
>
> Matrix multiplication is NOT commutative: $AB \neq BA$ in almost all cases, so you must preserve the order of matrices in all calculations.

**Worked example:** Given $A = \begin{pmatrix} 1 & 3 \\ 2 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & -1 \\ 5 & 2 \end{pmatrix}$, calculate $AB$.

1. Compute first row of A dotted with first column of B for top-left element: $1\times4 + 3\times5 = 19$
2. First row of A dotted with second column of B for top-right element: $1\times(-1) + 3\times2 = 5$
3. Second row of A dotted with first column of B for bottom-left element: $2\times4 + 0\times5 = 8$
4. Second row of A dotted with second column of B for bottom-right element: $2\times(-1) + 0\times2 = -2$
5. Combine into product matrix:

   $$AB = \begin{pmatrix} 19 & 5 \\ 8 & -2 \end{pmatrix}$$

> **Exam tip:** If a question asks for $BA$, do not reuse your $AB$ result, recalculate from scratch to avoid order errors.

*Calculator:* allowed

## 2×2 Determinants & Singular Matrices

**Determinant of a 2×2 matrix** — For $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\det M = ad - bc$. A matrix is singular if its determinant is 0, non-singular if determinant is non-zero.

*Notation:* $\det M$ or $|M|$

**Worked example:** For the matrix $M = \begin{pmatrix} 3 & k \\ 2 & 6 \end{pmatrix}$, find the value of $k$ for which $M$ is singular.

1. Write the determinant formula for M:

   $$\det M = (3)(6) - (k)(2) = 18 - 2k$$
2. Set determinant equal to 0 for singular matrix: $18 - 2k = 0$
3. Solve for k: $2k = 18 \implies k = 9$

> **Exam tip:** If you calculate a determinant of 0 for a matrix you need to invert, double check your arithmetic first, as singular matrices have no inverse.

*Calculator:* allowed

## Inverse of a 2×2 Matrix

The inverse of a non-singular 2×2 matrix $M$, written $M^{-1}$, is the matrix that satisfies $MM^{-1} = M^{-1}M = I$, where $I$ is the 2×2 identity matrix $\begin{pmatrix}1&0\\0&1\end{pmatrix}$. The formula for $M^{-1}$ is $\frac{1}{\det M} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.

**Worked example:** Find the inverse of $M = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$.

1. Calculate determinant of M:

   $$\det M = (4)(3) - (1)(2) = 12 - 2 = 10$$
2. Swap leading diagonal elements, change signs of off-diagonal elements to get adjugate matrix:

   $$\text{Adjugate} = \begin{pmatrix} 3 & -1 \\ -2 & 4 \end{pmatrix}$$
3. Multiply adjugate by $1/\det M$ to get inverse:

   $$M^{-1} = \frac{1}{10} \begin{pmatrix} 3 & -1 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} 0.3 & -0.1 \\ -0.2 & 0.4 \end{pmatrix}$$

> **Exam tip:** Always verify your inverse by multiplying it with the original matrix to check you get the identity matrix, if you have spare time in the exam.

*Calculator:* allowed

## Inverse of a Product of Matrices

For two non-singular 2×2 matrices $A$ and $B$, the inverse of their product follows the reverse order rule: $(AB)^{-1} = B^{-1}A^{-1}$. This rule extends to products of more than two matrices, with all inverses written in reverse order of the original product.

**Worked example:** Given $A^{-1} = \begin{pmatrix} 2 & 0 \\ -1 & 3 \end{pmatrix}$ and $B^{-1} = \begin{pmatrix} 1 & 4 \\ 0 & 2 \end{pmatrix}$, calculate $(AB)^{-1}$.

1. Apply reverse order inverse rule: $(AB)^{-1} = B^{-1}A^{-1}$
2. Calculate product of $B^{-1}$ and $A^{-1}$:

   $$B^{-1}A^{-1} = \begin{pmatrix} 1\times2 + 4\times(-1) & 1\times0 +4\times3 \\ 0\times2 + 2\times(-1) & 0\times0 + 2\times3 \end{pmatrix} = \begin{pmatrix} -2 & 12 \\ -2 & 6 \end{pmatrix}$$

> **Exam tip:** Never write $(AB)^{-1}$ as $A^{-1}B^{-1}$, this is the most common mistake on this topic in FP1 exams.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Adding matrices of different dimensions
  - Why it fails: Addition/subtraction is only defined for matrices with matching row and column counts
  - Correct: Check dimensions first before attempting addition or subtraction; if they don't match, the operation is undefined
- **Wrong:** Assuming $AB = BA$ for matrix multiplication
  - Why it fails: Matrix multiplication is non-commutative for almost all pairs of matrices
  - Correct: Preserve the order of matrices in all products, recalculate if asked for the reverse product
- **Wrong:** Writing $(AB)^{-1} = A^{-1}B^{-1}$
  - Why it fails: The product $A^{-1}B^{-1}$ does not cancel $AB$ to give the identity matrix
  - Correct: Always reverse the order of inverses: $(AB)^{-1} = B^{-1}A^{-1}$
- **Wrong:** Swapping all elements instead of just the leading diagonal when calculating the adjugate matrix
  - Why it fails: The adjugate matrix only requires a leading diagonal swap and sign change for off-diagonal elements
  - Correct: For $M = \begin{pmatrix}a&b\c&d\end{pmatrix}$, adjugate is $\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$
- **Wrong:** Attempting to find the inverse of a singular matrix
  - Why it fails: Singular matrices have determinant 0, so the inverse formula requires dividing by 0, which is undefined
  - Correct: Calculate determinant first before finding an inverse; if det=0, state the matrix is singular and has no inverse

## Cheatsheet

| Operation | Rule for 2×2 matrices $A = \begin{pmatrix}a&b\c&d\end{pmatrix}, B = \begin{pmatrix}e&f\g&h\end{pmatrix}$ |
| --- | --- |
| Addition/Subtraction | $A \pm B = \begin{pmatrix}a\pm e & b\pm f \\ c\pm g & d\pm h\end{pmatrix}$ (same dimensions only) |
| Scalar Multiplication | $kA = \begin{pmatrix}ka & kb \\ kc & kd\end{pmatrix}$ for scalar $k$ |
| Matrix Multiplication | $AB = \begin{pmatrix}ae+bg & af+bh \\ ce+dg & cf+dh\end{pmatrix}$ (conformable only) |
| Determinant | $\det A = ad - bc$ |
| Inverse | $A^{-1} = \frac{1}{ad-bc}\begin{pmatrix}d & -b \\ -c & a\end{pmatrix}$ (det $A \neq 0$ only) |
| Product Inverse | $(AB)^{-1} = B^{-1}A^{-1}$ |

## What's next

Now that you have mastered core 2x2 matrix algebra for FP1, you are ready to apply these skills to matrix transformations, the next key topic in the FP1 syllabus. Matrix operations are also foundational for later FP3 content including 3×3 matrices, eigenvalues and eigenvectors, and solving systems of linear equations. Make sure to practice past exam questions on this topic to avoid common order errors and memorise the determinant and inverse formulae, as they are not provided in the exam formula booklet.

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