Suppose A commutes with every 2 by 2 matrix (A B=B A), and...

Suppose $A$ commutes with every 2 by 2 matrix $(A B=B A)$, and in particular $$ A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \quad \text { commutes with } \quad B_{1}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right] \text { and } \quad B_{2}=\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right] $$ Show that $a=d$ and $b=c=0$. If $A B=B A$ for all matrices $B$, then $A$ is a multiple of the identity.

Suppose $A$ commutes with every 2 by 2 matrix $(A B=B A)$, and in particular
$$
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \quad \text { commutes with } \quad B_{1}=\left[\begin{array}{ll}
1 & 0 \\
0 & 0
\end{array}\right] \text { and } \quad B_{2}=\left[\begin{array}{ll}
0 & 1 \\
0 & 0
\end{array}\right]
$$
Show that $a=d$ and $b=c=0$. If $A B=B A$ for all matrices $B$, then $A$ is a multiple of the identity.

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Key Concepts

Matrix Commutation

This concept refers to the condition where two matrices A and B satisfy the relation AB = BA. Understanding commutation is critical in linear algebra since it reveals special algebraic properties about matrices and often simplifies computations. In the study of matrix algebra, the property of commutation is used to classify matrices and understand the structure of matrix algebras, particularly when determining cases such as the simultaneous diagonalization of matrices.

Center of a Matrix Algebra

The center of a matrix algebra consists of all matrices that commute with every other matrix in that algebra. This concept is pivotal because it identifies matrices that are 'universal commutants'. In many cases, including the one described in the problem, the center turns out to be the set of scalar matrices, those which are multiples of the identity matrix.

Scalar Matrices

A scalar matrix is a special kind of diagonal matrix in which every entry on the main diagonal is the same, meaning it is a multiple of the identity matrix. Scalar matrices automatically commute with any other matrix of compatible dimensions. This property is significant in various fields, especially in the context of characterizing matrices that commute with all other matrices in an algebra.

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