Matrix Multiplication

Find AB, where A = $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and B = $\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$.

• 1Top-left: ($1\times 5$) + ($2\times 7$) = $5+14=19$
• 2Top-right: ($1\times 6$) + ($2\times 8$) = $6+16=22$
• 3Bottom-left: ($3\times 5$) + ($4\times 7$) = $15+28=43$
• 4Bottom-right: ($3\times 6$) + ($4\times 8$) = $18+32=50$

Answer: AB = $\left[\begin{array}{cc}19& 22\\ 43& 50\end{array}\right]$

Using the same A and B, find BA, and compare with AB.

• 1Top-left: ($5\times 1$) + ($6\times 3$) = $5+18=23$
• 2Top-right: ($5\times 2$) + ($6\times 4$) = $10+24=34$
• 3Bottom-left: ($7\times 1$) + ($8\times 3$) = $7+24=31$
• 4Bottom-right: ($7\times 2$) + ($8\times 4$) = $14+32=46$

Answer: BA = $\left[\begin{array}{cc}23& 34\\ 31& 46\end{array}\right]$ — different from AB! Matrix multiplication is NOT commutative: order matters.

Find AB, where

• 1Columns of A ($3$) match rows of B ($3$), so AB exists and is $2\times 2$.
• 2Top-left: $(1\times 2)+(2\times 0)+(0\times 4)=2$
• 3Top-right: $(1\times 1)+(2\times 3)+(0\times 0)=7$
• 4Bottom-left: $(3\times 2)+(0\times 0)+(1\times 4)=10$
• 5Bottom-right: $(3\times 1)+(0\times 3)+(1\times 0)=3$

Answer: AB = $\left[\begin{array}{cc}2& 7\\ 10& 3\end{array}\right]$

Find the first row of CD, where

• 1Both are $3\times 3$, so CD is $3\times 3$.
• 2Row $1$ of C is $[1,0,2]$.
• 3Column $1$ of D: $(1\times 2)+(0\times 0)+(2\times 1)=4$
• 4Column $2$ of D: $(1\times 1)+(0\times 3)+(2\times 0)=1$
• 5Column $3$ of D: $(1\times 0)+(0\times 1)+(2\times 2)=4$

Answer: first row of CD = $[4,1,4]$