Zero Matrix (Null Matrix)

Zero Matrix, is a matrix where all the values are zero. It is denoted by subscript 0, like A₀. It is a special type of matrix and have some unique properties.

Example of a Zero Matrix

There is no restriction on size for being a zero matrix. Zero matrix don't need to be a square matrix. Here are some examples of zero matrix:

2 x 2 Zero Matrix:

\(A = \begin{bmatrix}
0 & 0 \\
0 & 0
\end{bmatrix}

\)

3 x 3 Zero Matrix:

\(A = \begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}

\)

Properties of Zero Matrix

Additive Identity:

Zero matrix is the additive identity for matrices. For any matrix A, A + 0 = A. where 0 is the zero matrix of the same size as A.

Scalar Multiplication:

For any scalar k, k * 0 = 0.

where 0 is the zero matrix of any size.

Is Zero Matrix Diagonalizable:

Yes, zero matrix is diagonalizable. But it should be a square matrix

Invertible:

The determinant of zero matrix is always zero. So, zero matrix is not invertible. For a matrix to be invertible, the determinant should be non-zero.

Rank:

Rank of zero matrix is always zero

Null Space:

Null space of zero matrix is the whole space.

Determinant:

Determinant of an matrix is calculated by the formula:

det(A) = Σ (-1)^i+j * a_ij * det(M_ij)

where M_ij is the matrix obtained by removing the i-th row and j-th column from A.

As all the elements in zero matrix are zero, the determinant of zero matrix is always zero.

Trace:

Trace is the sum of a matrix's diagonal elements. As all the elements in zero matrix are zero,

the trace of zero matrix is always zero.

Frequently Asked Questions (FAQs)

is the zero matrix diagonalizable?

Zero matrix (if it is a square matrix) is diagonalizable.

is zero matrix invertible?

Zero matrix (has a determinant of zero) is not invertible.

What is the symbol for zero matrix?

Zero matrix is denoted by subscript 0, like A₀.

what is the rank of zero matrix?

Rank of zero matrix is always zero.

What is the null space of zero matrix?

As all the elements in zero matrix are zero, the null space of zero matrix is the whole space.

What is the determinant of zero matrix?

The determinant of zero matrix is always zero.

What is the trace of zero matrix?

Trace (sum of diagonal elements) of zero matrix is always zero.

What is the transpose of zero matrix?

Transpose of matrix is interchanging rows and columns of matrix. Transposing the matrix only changes its dimension and not the values. So, transpose of zero matrix is also a zero matrix.

What is the adjoint of zero matrix?

Adjoint matrix is a transpose of the cofactor matrix. As zero matrix have all elements as zero,So the adjoint of zero matrix is a zero matrix.

What is the inverse of zero matrix?

Zero matrix don't have inverse

What is the eigenvalue of zero matrix?

Only square matrix have eigen values. If zero matrix is a square matrix, then the eigenvalue of zero matrix is zero.

What is the row space of zero matrix?

The row space of zero matrix is only the zero vector.

All rows of zero matrix are identical and equal to zero vector, therefore the row space of zero matrix is simply the zero vector.

What is the column space of zero matrix?

The column space of zero matrix is only the zero vector.

All columns of zero matrix are identical and equal to zero vector, therefore the column space of zero matrix is simply the zero vector.

What is the nullity of zero matrix?

The nullity of zero matrix is the number of columns in the matrix.

For example,

• The nullity of 2 x 2 zero matrix is 2.
• The nullity of 3 x 3 zero matrix is 3.
• The nullity of 2 x 3 zero matrix is 3.
• The nullity of 3 x 2 zero matrix is 2.