Question

Dadas as matrizes $A \ = \ \left[\begin{array}{ccc}2&-3\\1&4\\\end{array}\right]$ e $B \ = \ \left[\begin{array}{ccc}1&2\\-1&5\\\end{array}\right] ,$ determine as matrizes A · B e B · A:

Answer 1

Achando AB:

$A\cdot B=\left[\begin{array}{cc}2&-3\\1&4\end{array}\right]\cdot\left[\begin{array}{cc}1&2\\-1&5\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}(2\cdot1+[-3]\cdot[-1])&(2\cdot2+[-3]\cdot5)\\(1\cdot1+4\cdot[-1])&(1\cdot2+4\cdot5)\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}(2+3)&(4-15)\\(1-4)&(2+20)\end{array}\right]\\\\\\\boxed{\boxed{A\cdot B=\left[\begin{array}{cc}5&-11\\-3&~~22\end{array}\right]}}$

Achando BA:

$B\cdot A=\left[\begin{array}{cc}1&2\\-1&5\end{array}\right]\cdot\left[\begin{array}{cc}2&-3\\1&4\end{array}\right]\\\\\\B\cdot A=\left[\begin{array}{cc}(1\cdot2+2\cdot1)&(1\cdot[-3]+2\cdot4)\\([-1]\cdot2+5\cdot1)&([-1]\cdot[-3]+5\cdot4)\end{array}\right]\\\\\\B\cdot A=\left[\begin{array}{cc}(2+2)&(-3+8)\\(-2+5)&(3+20)\end{array}\right]\\\\\\\boxed{\boxed{B\cdot A=\left[\begin{array}{cc}4&5\\3&23\end{array}\right]}}$