Introduction to the Identity Matrix Revision Notes | GCSE AQA Higher Further Maths

1. What Is the Identity Matrix?

You already know that multiplying any number by 1 leaves it unchanged: $7 \times 1 = 7$. The identity matrix $I$ is the matrix equivalent of the number 1 — multiply any $2 \times 2$ matrix by $I$ and you get that matrix back, unaltered.

That single idea is the whole point of $I$. Before we see it in action, let's pin down exactly what it looks like.

Notice the pattern: 1s on the diagonal, 0s everywhere else. That specific arrangement is what gives $I$ its special property — and it's worth committing to memory, because exam questions can ask you to write it down directly.

2. Why $I$ Behaves Like the Number 1

The defining property of $I$ is this: for any $2 \times 2$ matrix $A$, multiplying by $I$ on either side returns $A$ exactly. In symbols: $AI = A$ and $IA = A$.

This is worth pausing on. Most matrix operations are sensitive to order — $AB$ and $BA$ are usually different. But $I$ is the exception: it commutes with every $2 \times 2$ matrix and always gives back the original.

Now let's verify this property by actually carrying out the multiplication row by column, so you can see precisely why it works rather than taking it on faith.

Worked example: Verifying $AI = A$ for a General $2 \times 2$ Matrix

Let $A = \begin{pmatrix} 3 & 5 \\ 1 & 4 \end{pmatrix}$ and $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Show that $AI = A$.

1. $AI = \begin{pmatrix} 3 & 5 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ — Write both matrices side by side ready for row-by-column multiplication. [M1]
2. Row 1 of $A$ $\cdot$ Column 1 of $I$: $(3 \times 1) + (5 \times 0) = 3 + 0 = 3$ — The 0 in column 1 of $I$ wipes out the second term, leaving the original entry 3.
3. Row 1 of $A$ $\cdot$ Column 2 of $I$: $(3 \times 0) + (5 \times 1) = 0 + 5 = 5$ — The 0 in column 2 of $I$ wipes out the first term, leaving the original entry 5.
4. Row 2 of $A$ $\cdot$ Column 1 of $I$: $(1 \times 1) + (4 \times 0) = 1 + 0 = 1$ — Same pattern: the 0 eliminates the unwanted term.
5. Row 2 of $A$ $\cdot$ Column 2 of $I$: $(1 \times 0) + (4 \times 1) = 0 + 4 = 4$ — Again, the 0 removes the first term and the 1 preserves the original entry.
6. $AI = \begin{pmatrix} 3 & 5 \\ 1 & 4 \end{pmatrix} = A$ ✓ — Every entry matches $A$ exactly, confirming $AI = A$. [A1]

Answer: $AI = \begin{pmatrix} 3 & 5 \\ 1 & 4 \end{pmatrix} = A$

Look at what happened in each step: the 1s in $I$ preserved each entry of $A$, and the 0s cancelled the entries that would otherwise have mixed things up. That's the structural reason $I$ works as an identity.

3. Multiplying $I$ on Both Sides

The worked example above showed $AI = A$. Now let's confirm the other direction — $IA = A$ — using the same matrix. This matters because matrix multiplication is generally not commutative, so both cases deserve explicit verification.

Worked example: Verifying $IA = A$

Using $A = \begin{pmatrix} 3 & 5 \\ 1 & 4 \end{pmatrix}$, calculate $IA$ and confirm it equals $A$.

1. $IA = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 3 & 5 \\ 1 & 4 \end{pmatrix}$ — This time $I$ is on the left. [M1]
2. Row 1 of $I$ $\cdot$ Column 1 of $A$: $(1 \times 3) + (0 \times 1) = 3 + 0 = 3$ — The 0 in row 1 of $I$ eliminates the second term.
3. Row 1 of $I$ $\cdot$ Column 2 of $A$: $(1 \times 5) + (0 \times 4) = 5 + 0 = 5$ — Again, the 0 removes the unwanted contribution.
4. Row 2 of $I$ $\cdot$ Column 1 of $A$: $(0 \times 3) + (1 \times 1) = 0 + 1 = 1$ — The 0 in row 2 of $I$ now eliminates the first term.
5. Row 2 of $I$ $\cdot$ Column 2 of $A$: $(0 \times 5) + (1 \times 4) = 0 + 4 = 4$ — The 1 preserves the entry; the 0 removes the other.
6. $IA = \begin{pmatrix} 3 & 5 \\ 1 & 4 \end{pmatrix} = A$ ✓ — Both $AI$ and $IA$ equal $A$, confirming $I$ is the multiplicative identity. [A1]

Answer: $IA = \begin{pmatrix} 3 & 5 \\ 1 & 4 \end{pmatrix} = A$

Both directions give the same result. This is a defining property of $I$: it commutes with every $2 \times 2$ matrix, returning the original matrix unchanged — a property captured precisely by $AI = IA = A$. Among all $2 \times 2$ matrices, $I$ is the unique multiplicative identity with this exact behaviour.

4. Formula Reference and Practice

Here is a compact reference for everything you need to recall about the identity matrix in an exam. Keep this in mind when a question asks you to write down $I$, use it in a product, or explain its effect.

5. Exam Technique and Common Mistakes

Questions on the identity matrix tend to be short — but they carry marks that are easy to drop through careless errors. The traps below are the ones that appear most often.

Mastering the identity matrix means knowing its exact form, understanding why $AI = IA = A$ holds, and being able to carry out the multiplication in full — these are the three things the specification requires, and the three things an examiner will look for.