Question

question 1-6
matrix $a = \

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

$ and matrix $b = \

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

$. stephanie concluded that $ab = ba$. is her conclusion valid?
her conclusion is valid because matrix multiplication is associative.
her conclusion is valid because $ab = \

$$\begin{bmatrix}4&3\\24&-2\\end{bmatrix}$$

$ and $ba = \

$$\begin{bmatrix}4&3\\24&-2\\end{bmatrix}$$

$.
her conclusion is not valid because matrix multiplication is not associative.
her conclusion is not valid because $ab = \

$$\begin{bmatrix}-8&4\\16&-8\\end{bmatrix}$$

$ and $ba = \

$$\begin{bmatrix}-2&-7\\-4&-14\\end{bmatrix}$$

$.

Explanation:

Step1: Recall matrix - multiplication formula

For two matrices $A=(a_{ij})$ of size $m\times n$ and $B=(b_{ij})$ of size $n\times p$, the $(i, j)$ - entry of the product $AB$ is given by $\sum_{k = 1}^{n}a_{ik}b_{kj}$.

Step2: Calculate $AB$

\[

$$\begin{align*} AB&=\begin{bmatrix}2&- 3\\0&1\end{bmatrix}\begin{bmatrix}2&-1\\4&-2\end{bmatrix}\\ &=\begin{bmatrix}2\times2+( - 3)\times4&2\times(-1)+(-3)\times(-2)\\0\times2 + 1\times4&0\times(-1)+1\times(-2)\end{bmatrix}\\ &=\begin{bmatrix}4 - 12&-2 + 6\\4&-2\end{bmatrix}=\begin{bmatrix}-8&4\\4&-2\end{bmatrix} \end{align*}$$

\]

Step3: Calculate $BA$

\[

$$\begin{align*} BA&=\begin{bmatrix}2&-1\\4&-2\end{bmatrix}\begin{bmatrix}2&-3\\0&1\end{bmatrix}\\ &=\begin{bmatrix}2\times2+( - 1)\times0&2\times(-3)+(-1)\times1\\4\times2+(-2)\times0&4\times(-3)+(-2)\times1\end{bmatrix}\\ &=\begin{bmatrix}4&-6 - 1\\8&-12 - 2\end{bmatrix}=\begin{bmatrix}4&-7\\8&-14\end{bmatrix} \end{align*}$$

\]
Since $AB
eq BA$, her conclusion is not valid.

Answer:

Her conclusion is not valid because $AB=

$$\begin{bmatrix}-8&4\\4&-2\end{bmatrix}$$

$ and $BA=

$$\begin{bmatrix}4&-7\\8&-14\end{bmatrix}$$

$.