AcademyMatrix Algebra
Academy
Identity And Zero Matrices
Level 1 - Linear Algebra topic page in Matrix Algebra.
Principle
The zero matrix is the additive neutral matrix: adding it changes nothing. The identity matrix is the multiplicative neutral square matrix: multiplying by it changes nothing when the product is defined.
These matrices play the same structural roles as \(0\) and \(1\) for real numbers, but their sizes must match the operation.
Notation
\(0_{m,n}\)
the m by n zero matrix
\(I_n\)
the n by n identity matrix
\(\delta_{ij}\)
Kronecker delta, equal to 1 when i=j and 0 otherwise
\(A\)
a matrix whose size determines which zero or identity matrix is needed
The Core Method
The zero matrix has every entry equal to zero:
Zero matrix
\[(0_{m,n})_{ij}=0\]
The identity matrix has ones on the main diagonal and zeros elsewhere:
Identity entries
\[(I_n)_{ij}=\delta_{ij}\]
For an \(m\) by \(n\) matrix \(A\),
Additive identity
\[A+0_{m,n}=A\]
and
Multiplicative identities
\[I_mA=A,\qquad AI_n=A\]
The identity size changes depending on the side of multiplication.
Worked Cases
Question
Let \(A\) be a \(3\) by \(2\) matrix. What are the sizes of the identity matrices in \(I_mA=A\) and \(AI_n=A\)?
Answer
Since \(A\) has \(3\) rows and \(2\) columns, left multiplication needs \(I_3\): \(I_3A=A\). Right multiplication needs \(I_2\): \(AI_2=A\). The identity matrices have different sizes because they act on different sides.
Examples
Question
Write \(I_4\).
Answer
The matrix \(I_4\) is \(4\) by \(4\), with ones on the main diagonal and zeros elsewhere: \(I_4=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}\).