Scalar Multiples, Transposes, Sums, and Products of Partitioned Matrices

Definition

Let

$$A=\left[\begin{array}{cccc}{A}_{11}& A_{12}& \dots & A_{1c}\\ A_{21}& A_{22}& \dots & A_{2c}\\ ⋮& ⋮& \ddots & ⋮\\ A_{r1}& A_{r2}& \dots & A_{rc}\end{array}\right]$$

represent a partitioned $m\times n$ matrix whose $ij$-th block $A_{ij}$ is of dimensions $m_i\times n_j$.

Then