In mathematics, a Redheffer matrix, often denoted ${\displaystyle A_{n}}$ as studied by Redheffer (1977), is a square (0,1) matrix whose entries a_ij are 1 if i divides j or if j = 1; otherwise, a_ij = 0. It is useful in some contexts to express Dirichlet convolution, or convolved divisors sums, in terms of matrix products involving the transpose of the ${\displaystyle n^{th}}$ Redheffer matrix.

Since the invertibility of the Redheffer matrices are complicated by the initial column of ones in the matrix, it is often convenient to express ${\displaystyle A_{n}:=C_{n}+D_{n}}$ where ${\displaystyle C_{n}:=[c_{ij}]}$ is defined to be the (0,1) matrix whose entries are one if and only if ${\displaystyle j=1}$ and ${\displaystyle i\neq 1}$. The remaining one-valued entries in ${\displaystyle A_{n}}$ then correspond to the divisibility condition reflected by the matrix ${\displaystyle D_{n}}$, which plainly can be seen by an application of Mobius inversion is always invertible with inverse ${\displaystyle D_{n}^{-1}=\left[\mu (j/i)M_{i}(j)\right]}$. We then have a characterization of the singularity of ${\displaystyle A_{n}}$ expressed by ${\displaystyle \det \left(A_{n}\right)=\det \left(D_{n}^{-1}C_{n}+I_{n}\right).}$

If we define the function

then we can define the ${\displaystyle n^{th}}$ Redheffer (transpose) matrix to be the nxn square matrix ${\displaystyle R_{n}=[M_{j}(i)]_{1\leq i,j\leq n}}$ in usual matrix notation. We will continue to make use this notation throughout the next sections.

The matrix below is the 12 × 12 Redheffer matrix. In the split sum-of-matrices notation for ${\displaystyle A_{12}:=C_{12}+D_{12}}$, the entries below corresponding to the initial column of ones in ${\displaystyle C_{n}}$ are marked in blue.

A corresponding application of the Mobius inversion formula shows that the ${\displaystyle n^{th}}$ Redheffer transpose matrix is always invertible, with inverse entries given by

where ${\displaystyle \mu (n)}$ denotes the Moebius function. In this case, we have that the ${\displaystyle 12\times 12}$ inverse Redheffer transpose matrix is given by

The determinant of the n × n square Redheffer matrix is given by the Mertens function M(n). In particular, the matrix ${\displaystyle A_{n}}$ is not invertible precisely when the Mertens function is zero (or is close to changing signs). As a corollary of the disproof of the Mertens conjecture, it follows that the Mertens function changes sign, and is therefore zero, infinitely many times, so the Redheffer matrix ${\displaystyle A_{n}}$ is singular at infinitely many natural numbers.