Solved Examples: Block Matrix Multiplication and Inversion Detailed

📚 Quick Study Guide

• 🔢 Block Matrix Multiplication: If $A$ and $B$ are partitioned matrices such that the column partitioning of $A$ matches the row partitioning of $B$, then the product $AB$ can be computed by multiplying the blocks as if they were scalars, provided the block sizes are compatible for matrix multiplication.
• ➕ The $(i,j)$-th block of $AB$ is given by $\sum_{k=1}^{p} A_{ik}B_{kj}$, where $A$ is partitioned into $m \times p$ blocks and $B$ into $p \times n$ blocks.
• 🔄 Block Matrix Inversion (2x2 Case): For a $2 \times 2$ block matrix $M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$, the inverse $M^{-1}$ can be found as follows (assuming all inverses exist):
  • If $A$ is invertible, $M^{-1} = \begin{bmatrix} A^{-1} + A^{-1}B(D - CA^{-1}B)^{-1}CA^{-1} & -A^{-1}B(D - CA^{-1}B)^{-1} \\ -(D - CA^{-1}B)^{-1}CA^{-1} & (D - CA^{-1}B)^{-1} \end{bmatrix}$.
  • If $D$ is invertible, $M^{-1} = \begin{bmatrix} (A - BD^{-1}C)^{-1} & - (A - BD^{-1}C)^{-1}BD^{-1} \\ -D^{-1}C(A - BD^{-1}C)^{-1} & D^{-1} + D^{-1}C(A - BD^{-1}C)^{-1}BD^{-1} \end{bmatrix}$.
• 💡 Schur Complement: The expressions $D - CA^{-1}B$ and $A - BD^{-1}C$ are known as Schur complements. These are essential for block matrix inversion.

Practice Quiz

1. Question 1: Given $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, consider them as blocks. If $M = \begin{bmatrix} A & B \\ B & A \end{bmatrix}$, what is the (1,1) block of $M^2$?
   1. $A^2 + B^2$
   2. $2AB$
   3. $A^2 + 2AB + B^2$
   4. $AB + BA$
2. Question 2: Let $M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$. Assuming $A$ is invertible, what is the Schur complement of $A$ in $M$?
   1. $D - CA^{-1}B$
   2. $D + CA^{-1}B$
   3. $A - BD^{-1}C$
   4. $A + BD^{-1}C$
3. Question 3: If $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, $C = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$, and $D = \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}$, what is the result of $AB$?
   1. $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
   2. $\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$
   3. $\begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}$
   4. $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
4. Question 4: Consider the block matrix $M = \begin{bmatrix} A & 0 \\ 0 & D \end{bmatrix}$, where $A$ and $D$ are invertible. What is $M^{-1}$?
   1. $\begin{bmatrix} A^{-1} & 0 \\ 0 & D^{-1} \end{bmatrix}$
   2. $\begin{bmatrix} D^{-1} & 0 \\ 0 & A^{-1} \end{bmatrix}$
   3. $\begin{bmatrix} 0 & A^{-1} \\ D^{-1} & 0 \end{bmatrix}$
   4. $\begin{bmatrix} 0 & D^{-1} \\ A^{-1} & 0 \end{bmatrix}$
5. Question 5: Let $M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$. If $A$ is invertible, what block is in the (2,1) position of $M^{-1}$?
   1. $-(D - CA^{-1}B)^{-1}CA^{-1}$
   2. $-A^{-1}B(D - CA^{-1}B)^{-1}$
   3. $(D - CA^{-1}B)^{-1}$
   4. $A^{-1} + A^{-1}B(D - CA^{-1}B)^{-1}CA^{-1}$
6. Question 6: If $A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$, $C = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$, and $D = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, and $M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$, what is $CA^{-1}B$?
   1. $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
   2. $\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$
   3. $\begin{bmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{bmatrix}$
   4. $\begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}$
7. Question 7: For a block matrix $M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$, if $A$ and $(D - CA^{-1}B)$ are invertible, can $M$ be invertible?
   1. Yes
   2. No
   3. Only if $B$ and $C$ are zero matrices
   4. Only if $A = D$