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Zero Matrix — Definition, Properties & Examples

Zero Matrix

A matrix for which all elements are equal to 0. Note: The zero matrix is the identity for matrix addition.

 

3×2 zero matrix with all six elements equal to 0, arranged in 3 rows and 2 columns of zeros.

 

 

See also

Dimensions of a matrix

Key Formula

Om×n=[00⋯000⋯0⋮⋮⋱⋮00⋯0]O_{m \times n} = \begin{bmatrix} 0 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{bmatrix}
Where:
  • Om×nO_{m \times n} = The zero matrix with m rows and n columns
  • mm = The number of rows in the matrix
  • nn = The number of columns in the matrix

Worked Example

Problem: Let A be the 2×3 matrix shown below. Verify that A + O = A, where O is the 2×3 zero matrix. A = [[4, -1, 7], [3, 5, -2]]
Step 1: Write the 2×3 zero matrix. Every entry is 0.
O=[000000]O = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}
Step 2: Set up the addition A + O by adding corresponding elements.
A+O=[4−1735−2]+[000000]A + O = \begin{bmatrix} 4 & -1 & 7 \\ 3 & 5 & -2 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}
Step 3: Add each pair of corresponding entries: 4+0, −1+0, 7+0, 3+0, 5+0, −2+0.
A+O=[4+0−1+07+03+05+0−2+0]A + O = \begin{bmatrix} 4+0 & -1+0 & 7+0 \\ 3+0 & 5+0 & -2+0 \end{bmatrix}
Step 4: Simplify to get the result.
A+O=[4−1735−2]=AA + O = \begin{bmatrix} 4 & -1 & 7 \\ 3 & 5 & -2 \end{bmatrix} = A
Answer: A + O = A, confirming that the zero matrix is the additive identity for matrix addition.

Another Example

This example demonstrates multiplication rather than addition with the zero matrix, showing that the zero matrix acts as an absorbing element under matrix multiplication — analogous to how multiplying any number by 0 gives 0.

Problem: Multiply the 2×2 matrix B by the 2×2 zero matrix O. What is B · O?
Step 1: Define B and the 2×2 zero matrix.
B=[36−12],O=[0000]B = \begin{bmatrix} 3 & 6 \\ -1 & 2 \end{bmatrix}, \quad O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
Step 2: Compute the product B · O using row-by-column multiplication. For the (1,1) entry:
(B⋅O)1,1=3⋅0+6⋅0=0(B \cdot O)_{1,1} = 3 \cdot 0 + 6 \cdot 0 = 0
Step 3: Every entry follows the same pattern — each term in the dot product contains a factor of 0, so every entry of the result is 0.
B⋅O=[0000]=OB \cdot O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = O
Answer: B · O = O. Multiplying any matrix by a zero matrix (of compatible dimensions) always produces a zero matrix.

Frequently Asked Questions

What is the difference between a zero matrix and an identity matrix?
A zero matrix has every element equal to 0 and serves as the additive identity: A + O = A. An identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere, and it serves as the multiplicative identity: A · I = A. They play analogous roles but for different operations.
Can a zero matrix be any size?
Yes. A zero matrix can have any dimensions m × n. There is a 1×1 zero matrix [0], a 2×3 zero matrix, a 5×5 zero matrix, and so on. The key requirement is simply that every entry equals 0.
Is the zero matrix invertible?
No. A square zero matrix has a determinant of 0, which means it is singular and has no inverse. You cannot find a matrix that, when multiplied by the zero matrix, produces the identity matrix.

Zero Matrix vs. Identity Matrix

Zero MatrixIdentity Matrix
DefinitionAll elements are 01s on the main diagonal, 0s elsewhere
NotationO (sometimes 0 with subscript)I or Iₙ
DimensionsAny m × nMust be square (n × n)
Role in additionAdditive identity: A + O = ANo special role
Role in multiplicationAbsorbing element: A · O = OMultiplicative identity: A · I = A
Determinant (square case)Always 0Always 1
Invertible?NoYes

Why It Matters

You encounter the zero matrix whenever you study systems of linear equations, linear transformations, or vector spaces. It defines what "zero" means in matrix algebra, much as 0 defines it for ordinary numbers. Understanding the zero matrix is essential for proving properties like A − A = O and for recognizing when a system of equations has a trivial solution.

Common Mistakes

Mistake: Confusing the zero matrix with the identity matrix.
Correction: The zero matrix (all 0s) is the additive identity: A + O = A. The identity matrix (1s on the diagonal) is the multiplicative identity: A · I = A. They serve different roles for different operations.
Mistake: Assuming that if A · B = O, then either A = O or B = O.
Correction: Unlike ordinary numbers, two nonzero matrices can multiply to give the zero matrix. For example, A = [[1, 0], [0, 0]] and B = [[0, 0], [0, 1]] satisfy A · B = O even though neither A nor B is the zero matrix. This is called the existence of zero divisors.

Related Terms