Question

if $a = \

$$\begin{bmatrix} 2 & 0 & -1 \\\\ 3 & 5 & 2 \\\\ -4 & 1 & 4 \\end{bmatrix}$$

$ and $b = \

$$\begin{bmatrix} 5 & 1 & -2 \\\\ -1 & 0 & 4 \\\\ 2 & -3 & 3 \\end{bmatrix}$$

$, then what is $ab$?

Explanation:

Step1: Define matrix multiplication rule

For matrices $A_{m\times n}$ and $B_{n\times p}$, the element $C_{ij}$ of $AB=C$ is $\sum_{k=1}^n A_{ik}B_{kj}$.

Step2: Calculate first row of AB

Compute each element by dot product:
First element: $2\times5 + 0\times(-1) + (-1)\times2 = 10+0-2=8$
Second element: $2\times1 + 0\times0 + (-1)\times(-3) = 2+0+3=5$
Third element: $2\times(-2) + 0\times4 + (-1)\times3 = -4+0-3=-7$

Step3: Calculate second row of AB

Compute each element by dot product:
First element: $3\times5 + 5\times(-1) + 2\times2 = 15-5+4=14$
Second element: $3\times1 + 5\times0 + 2\times(-3) = 3+0-6=-3$
Third element: $3\times(-2) + 5\times4 + 2\times3 = -6+20+6=20$

Step4: Calculate third row of AB

Compute each element by dot product:
First element: $-4\times5 + 1\times(-1) + 4\times2 = -20-1+8=-13$
Second element: $-4\times1 + 1\times0 + 4\times(-3) = -4+0-12=-16$
Third element: $-4\times(-2) + 1\times4 + 4\times3 = 8+4+12=24$

Answer:

$$\begin{bmatrix} 8 & 5 & -7 \\ 14 & -3 & 20 \\ -13 & -16 & 24 \end{bmatrix}$$