16. determine if the matrices are inverses of each other by multiplying the matrices. $m = \\begin{bmatrix}…
16. determine if the matrices are inverses of each other by multiplying the matrices. $m = \\begin{bmatrix} 1 & 7 \\\\ 2 & 3 \\end{bmatrix}$ and $n = \\begin{bmatrix} 0 & 1 \\\\ 3 & -2 \\end{bmatrix}$\nyes\nno
Answer
Explanation:
Step1: Define matrix multiplication rule
For two 2x2 matrices $A=\begin{bmatrix}a&b\c&d\end{bmatrix}$ and $B=\begin{bmatrix}e&f\g&h\end{bmatrix}$, their product is: $$AB=\begin{bmatrix}ae+bg&af+bh\ce+dg&cf+dh\end{bmatrix}$$
Step2: Calculate $M \times N$
Substitute $M=\begin{bmatrix}1&7\2&3\end{bmatrix}$ and $N=\begin{bmatrix}0&1\3&-2\end{bmatrix}$: $$ \begin{align*} M\times N&=\begin{bmatrix}(1\times0)+(7\times3)&(1\times1)+(7\times-2)\(2\times0)+(3\times3)&(2\times1)+(3\times-2)\end{bmatrix}\ &=\begin{bmatrix}0+21&1-14\0+9&2-6\end{bmatrix}\ &=\begin{bmatrix}21&-13\9&-4\end{bmatrix} \end{align*} $$
Step3: Compare to identity matrix
The 2x2 identity matrix is $I=\begin{bmatrix}1&0\0&1\end{bmatrix}$. Since $M\times N \neq I$, the matrices are not inverses.
Answer:
no