Q.Verify the property $A(B+C)=AB+AC$, when the matrices $A, B,$ and $C$ are given by

[!FORMULA] $A=\begin{pmatrix} 2 & 0 & -3 \\ 1 & 4 & 5\end{pmatrix},\quad B=\begin{pmatrix} 3 & 1 \\ -1 & 0 \\ 4 & 2\end{pmatrix},\quad C=\begin{pmatrix} 4 & 7 \\ 2 & 1 \\ 1 & -1\end{pmatrix}$

Step 1. Compute $B+C$.

Step 2. Compute $A(B+C)$.

$(1,1):\ 2(7)+0(1)+(-3)(5)=14+0-15=-1$

$(1,2):\ 2(8)+0(1)+(-3)(1)=16+0-3=13$

$(2,1):\ 1(7)+4(1)+5(5)=7+4+25=36$

$(2,2):\ 1(8)+4(1)+5(1)=8+4+5=17$

So $A(B+C)=\begin{pmatrix}-1&13\\36&17\end{pmatrix}$.

Step 3. Compute $AB$ separately.

$(1,1):\ 2(3)+0(-1)+(-3)(4)=6+0-12=-6$

$(1,2):\ 2(1)+0(0)+(-3)(2)=2+0-6=-4$

$(2,1):\ 1(3)+4(-1)+5(4)=3-4+20=19$

$(2,2):\ 1(1)+4(0)+5(2)=1+0+10=11$

$AB=\begin{pmatrix}-6&-4\\19&11\end{pmatrix}$

Step 4. Compute $AC$ separately.

$(1,1):\ 2(4)+0(2)+(-3)(1)=8+0-3=5$

$(1,2):\ 2(7)+0(1)+(-3)(-1)=14+0+3=17$ …

• Adding $B$ and $C$ incorrectly before multiplying by $A$ …