Question

find the product of the following two matrices.
$\begin{bmatrix} 0&-1\\ -4&4\\ 0&5end{bmatrix} \begin{bmatrix} 1&-3&0&2\\ -1&0&2&1end{bmatrix}$

Explanation:

Step1: Define matrix product rule

For matrices $A_{m\times n}$ and $B_{n\times p}$, their product $C_{m\times p}$ has entries $C_{ij}=\sum_{k=1}^n A_{ik}B_{kj}$.

Step2: Calculate first row of product

Compute each entry using row 1 of first matrix:
$C_{11}=(0)(1)+(-1)(-1)=1$
$C_{12}=(0)(-3)+(-1)(0)=0$
$C_{13}=(0)(0)+(-1)(2)=-2$
$C_{14}=(0)(2)+(-1)(1)=-1$

Step3: Calculate second row of product

Compute each entry using row 2 of first matrix:
$C_{21}=(-4)(1)+(4)(-1)=-4-4=-8$
$C_{22}=(-4)(-3)+(4)(0)=12+0=12$
$C_{23}=(-4)(0)+(4)(2)=0+8=8$
$C_{24}=(-4)(2)+(4)(1)=-8+4=-4$

Step4: Calculate third row of product

Compute each entry using row 3 of first matrix:
$C_{31}=(0)(1)+(5)(-1)=0-5=-5$
$C_{32}=(0)(-3)+(5)(0)=0+0=0$
$C_{33}=(0)(0)+(5)(2)=0+10=10$
$C_{34}=(0)(2)+(5)(1)=0+5=5$

Answer:

$$\begin{bmatrix} 1 & 0 & -2 & -1 \\ -8 & 12 & 8 & -4 \\ -5 & 0 & 10 & 5 \end{bmatrix}$$