Matrix algebra

Q: When is matrix multiplication defined?

Matrix multiplication is only defined if the number of columns in equals the number of rows in . For FP1, all products you will encounter are between conformable 2×2 matrices.

Edexcel International A-Level Further Mathematics· FP1 5.1-5.5· 25 min read

1. Matrix Addition, Subtraction & Scalar Multiplication★☆☆☆☆⏱ 10 min

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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.

📘 Definition

Scalar multiplication

Multiplying every element of a matrix by a constant (scalar) value.

📐 Worked Example

Given $A = (2 - 3 14)$ and $B = (50 - 2 7)$ , calculate $2 A - B$ .

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.

2. Matrix Multiplication★★☆☆☆⏱ 15 min

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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 $A = (a c b d)$ and $B = (e g f h)$ , the product $A B = (a e + b g ce + d g a f + bh c f + d h)$ .

📐 Worked Example

Given $A = (1230)$ and $B = (45 - 1 2)$ , calculate $A B$ .

1
Compute first row of A dotted with first column of B for top-left element: $1 \times 4 + 3 \times 5 = 19$
2
First row of A dotted with second column of B for top-right element: $1 \times (- 1) + 3 \times 2 = 5$
3
Second row of A dotted with first column of B for bottom-left element: $2 \times 4 + 0 \times 5 = 8$
4
Second row of A dotted with second column of B for bottom-right element: $2 \times (- 1) + 0 \times 2 = - 2$
5
Combine into product matrix:
$AB = \begin{pmatrix} 19 & 5 \\ 8 & -2 \end{pmatrix}$

Exam tip:

If a question asks for $B A$ , do not reuse your $A B$ result, recalculate from scratch to avoid order errors.

3. 2×2 Determinants & Singular Matrices★★☆☆☆⏱ 10 min

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📘 Definition

Determinant of a 2×2 matrix

$det M$ or $∣ M ∣$

For $M = (a c b d)$ , $det M = a d - b c$ . A matrix is singular if its determinant is 0, non-singular if determinant is non-zero.

📐 Worked Example

For the matrix $M = (32 k 6)$ , 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: $18 - 2 k = 0$
3
Solve for k: $2 k = 18 ⟹ k = 9$

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.

4. Inverse of a 2×2 Matrix★★★☆☆⏱ 15 min

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The inverse of a non-singular 2×2 matrix $M$ , written $M^{- 1}$ , is the matrix that satisfies $M M^{- 1} = M^{- 1} M = I$ , where $I$ is the 2×2 identity matrix $(1001)$ . The formula for $M^{- 1}$ is $\frac{1}{d e t M} (d - c - b a)$ .

📐 Worked Example

Find the inverse of $M = (4213)$ .

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.

5. Inverse of a Product of Matrices★★★☆☆⏱ 10 min

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

📐 Worked Example

Given $A^{- 1} = (2 - 1 03)$ and $B^{- 1} = (1042)$ , calculate $(A B)^{- 1}$ .

1
Apply reverse order inverse rule: $(A B)^{- 1} = B^{- 1} A^{- 1}$
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 $(A B)^{- 1}$ as $A^{- 1} B^{- 1}$ , this is the most common mistake on this topic in FP1 exams.

6. Common Pitfalls

Wrong move:

Adding matrices of different dimensions

Why:

Addition/subtraction is only defined for matrices with matching row and column counts

Correct move:

Check dimensions first before attempting addition or subtraction; if they don't match, the operation is undefined

Wrong move:

Assuming $A B = B A$ for matrix multiplication

Why:

Matrix multiplication is non-commutative for almost all pairs of matrices

Correct move:

Preserve the order of matrices in all products, recalculate if asked for the reverse product

Wrong move:

Writing $(A B)^{- 1} = A^{- 1} B^{- 1}$

Why:

The product $A^{- 1} B^{- 1}$ does not cancel $A B$ to give the identity matrix

Correct move:

Always reverse the order of inverses: $(A B)^{- 1} = B^{- 1} A^{- 1}$

Wrong move:

Swapping all elements instead of just the leading diagonal when calculating the adjugate matrix

Why:

The adjugate matrix only requires a leading diagonal swap and sign change for off-diagonal elements

Correct move:

For $M = \begin{pmatrix}a&b\c&d\end{pmatrix}$ , adjugate is $(d - c - b a)$

Wrong move:

Attempting to find the inverse of a singular matrix

Why:

Singular matrices have determinant 0, so the inverse formula requires dividing by 0, which is undefined

Correct move:

Calculate determinant first before finding an inverse; if det=0, state the matrix is singular and has no inverse

7. Quick Reference 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	$A \pm B = (a \pm e c \pm g b \pm f d \pm h)$ (same dimensions only)
Scalar Multiplication	$k A = (k a k c k b k d)$ for scalar $k$
Matrix Multiplication	$A B = (a e + b g ce + d g a f + bh c f + d h)$ (conformable only)
Determinant	$det A = a d - b c$
Inverse	$A^{- 1} = \frac{1}{a d - b c} (d - c - b a)$ (det $A \neq = 0$ only)
Product Inverse	$(A B)^{- 1} = B^{- 1} A^{- 1}$

8. Frequently Asked

When is matrix multiplication defined?

Matrix multiplication $A B$ is only defined if the number of columns in $A$ equals the number of rows in $B$ . For FP1, all products you will encounter are between conformable 2×2 matrices.

Do I get the 2×2 inverse formula in the FP1 exam?

No, the 2×2 determinant and inverse formulae are not provided in the FP1 formula booklet, so you must memorise them for the exam.

Why is $(A B)^{- 1}$ not equal to $A^{- 1} B^{- 1}$ ?

Matrix multiplication is not commutative, so the order of inverses must be reversed to cancel out the original product correctly: $(A B) (B^{- 1} A^{- 1}) = A (B B^{- 1}) A^{- 1} = A I A^{- 1} = I$ , the identity matrix.

Going deeper

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.