Recipe

Multiply two block-partitioned matrices

Use when: you have two matrices split into blocks with matching block-column widths and block-row heights, and need one or more blocks of the product $C=AB$

Goal: the requested block (or blocks) of $C$ , computed by the same row-times-column rule as ordinary multiplication, applied one level up to sub-matrices

Taught in Block (partitioned) matrices

Quick recipe

Check that the partitions are conformable
Identify exactly which block products a target block needs
Compute each needed block product with ordinary matrix multiplication
Add the block products together entry-by-entry

1
$A$ 's block-column widths must match $B$ 's block-row heights — the same inner-dimension rule ordinary matrix multiplication always requires, just applied to whole groups of columns and rows instead of individual ones.

Example

$A$ is partitioned into $2\times2$ blocks with block-columns of width $2$ and $2$ ; $B$ is partitioned into block-rows of height $2$ and $2$ . The widths and heights line up, so $A_{it}B_{tj}$ will make sense for every $i,j,t$ .

Why: if a block's width didn't match its partner's height, the individual matrix product $A_{it}B_{tj}$ inside the block sum wouldn't be defined at all — conformability has to hold block by block, not just for the outer dimensions.
2
Block $(i,j)$ of $C$ is $\sum_t A_{it}B_{tj}$ — write out which specific pairs of blocks that sum actually calls for before multiplying anything.

Example

For $C_{12}$ (block-row $1$ , block-column $2$ ) with a $2\times2$ grid of blocks, $t$ ranges over $1,2$ : $C_{12}=A_{11}B_{12}+A_{12}B_{22}$ . Blocks $A_{21}$ , $A_{22}$ 's pairing with $B_{11}$ , and $B_{21}$ never enter this particular sum.

Why: this mirrors $(AB)_{ij}=\sum_t a_{it}b_{tj}$ exactly, one level up — knowing the needed pairs in advance means never wasting effort on a block that doesn't contribute to the one you were actually asked for.
3
Each $A_{it}B_{tj}$ is an ordinary matrix product — multiply it out row by row exactly as any other matrix multiplication.

Example

$A_{11}B_{12}=\begin{pmatrix}1&0\\2&1\end{pmatrix}\begin{pmatrix}1&2\\0&1\end{pmatrix}=\begin{pmatrix}1&2\\2&5\end{pmatrix}, \qquad A_{12}B_{22}=\begin{pmatrix}3&-1\\0&2\end{pmatrix}\begin{pmatrix}1&0\\-2&1\end{pmatrix}=\begin{pmatrix}5&-1\\-4&2\end{pmatrix}$

Why: a block is a whole sub-matrix, so "multiplying" it means the full matrix-multiplication recipe, not a single scalar multiplication — nothing about being a block changes how a matrix product itself is computed.
4
Sum the block products from Step 3 the way block addition always works: matching position, matching size, added entry by entry.

Example

$C_{12}=\begin{pmatrix}1&2\\2&5\end{pmatrix}+\begin{pmatrix}5&-1\\-4&2\end{pmatrix}=\begin{pmatrix}6&1\\-2&7\end{pmatrix}$

Why: the sum $\sum_t A_{it}B_{tj}$ is literally an ordinary matrix sum of the individual block products — nothing in the block picture changes what "add two matrices" means.

Common mistakes

Skipping the conformability check and multiplying two blocks whose inner sizes don't actually match.
Computing every block of the product when only one was asked for, instead of using Step 2 to find just the pairs a target block needs.
Dropping a term from the sum $\sum_t A_{it}B_{tj}$ — every value of $t$ contributes one block product, not just the first.
Adding the block products from Step 3 in the wrong order or against the wrong partner, mixing up which block product belongs to which target.
Treating a block like a single number — a $2\times2$ block still needs the full row-times-column recipe, not one multiplication.