Pre-Calculus
Matrices
Identity and zero matrices
20 practice questions
0 video lessons
Theory + worked examples
Theory
Two special matrices behave like the numbers \(1\) and \(0\):
- Identity matrix \(I\): square, with \(1\)s on the main diagonal and \(0\)s elsewhere. It satisfies \(AI=IA=A\).
- Zero matrix \(O\): all entries \(0\). It satisfies \(A+O=A\) and \(AO=O\).
The identity is the key to inverses. An inverse \(A^{-1}\) is defined by \(AA^{-1}=I\).
The \(2\times 2\) identity and zero matrices.
Their defining roles.
The identity and zero properties:
\[AI=IA=A,\qquad A+O=A,\qquad AO=OA=O\]
Only square matrices have an identity of the matching size.
How to use these matrices
- Identity: multiply by \(I\) to leave a matrix unchanged.
- Zero: add \(O\) to leave a matrix unchanged.
- Build \(I_n\): place \(1\)s down the diagonal, \(0\)s elsewhere.
- Recognize \(AA^{-1}=I\) when defining inverses.
Example 1 — Multiply by the identity
Compute \(\begin{bmatrix}3&5\\2&7\end{bmatrix}\begin{bmatrix}1&0\\0&1\end{bmatrix}\).
Solution
The identity leaves the matrix unchanged.
\[=\begin{bmatrix}3&5\\2&7\end{bmatrix}\]
Example 2 — Add the zero matrix
Compute \(\begin{bmatrix}4&-1\\0&6\end{bmatrix}+\begin{bmatrix}0&0\\0&0\end{bmatrix}\).
Solution
Adding the zero matrix changes nothing.
\[=\begin{bmatrix}4&-1\\0&6\end{bmatrix}\]
Example 3 — The 3×3 identity
Write the \(3\times 3\) identity matrix.
Solution
Ones on the main diagonal, zeros elsewhere.
\[I_3=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\]
Example 4 — Why the identity matters
What role does \(I\) play like the number 1?
Solution
Just as \(1\cdot x=x\), the identity satisfies \(AI=IA=A\) — it is the multiplicative identity for matrices, central to defining inverses.
| \(AI\) | \(=\) | \(A\) |
Common pitfalls
The identity is square. There is a different \(I_n\) for each size.
\(I\) has \(1\)s only on the main diagonal, not everywhere.
The zero matrix isn't the number 0. It must match the size for addition.
Frequently asked questions
What is the identity matrix?
A square matrix with \(1\)s on the main diagonal and \(0\)s elsewhere; it satisfies \(AI=IA=A\).
What is the zero matrix?
A matrix of all zeros; adding it leaves any matrix unchanged, and it acts like 0 under addition.
Why is the identity matrix important?
It is the multiplicative identity, and inverses are defined by \(AA^{-1}=I\).
Does every matrix have an identity?
The identity is square; each size \(n\) has its own \(I_n\) that works with \(n\times n\) matrices.
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