Question

Find AB if

a)$\mathit{A}\mathbf{=}\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]\mathbf{;}\mathit{B}\mathbf{=}\left[\begin{array}{cc}0& 4\\ 1& 3\end{array}\right]$

b)$\mathit{A}\mathbf{=}\left[\begin{array}{cc}1& -1\\ 0& 1\\ 2& 3\end{array}\right]\mathbf{;}\mathit{B}\mathbf{=}\left[\begin{array}{ccc}3& -2& -1\\ 1& 0& 2\end{array}\right]$

c)$\mathit{A}\mathbf{=}\left[\begin{array}{cc}4& -3\\ 3& -1\\ 0& -2\\ -1& 5\end{array}\right]\mathbf{;}\mathit{B}\mathbf{=}\left[\begin{array}{cccc}-1& 3& 2& -2\\ 0& -1& 4& -3\end{array}\right]$

Short answer

a) Required matrix is $\left[\begin{array}{cc}1& 11\\ 2& 18\end{array}\right]$.

b) Required matrix is $\left[\begin{array}{ccc}2& -2& -3\\ 1& 0& 3\\ 9& -4& 4\end{array}\right]$.

c) Required matrix is $\left[\begin{array}{cccc}4& 15& -4& 1\\ -3& 10& 2& 9\\ 0& 2& -8& 6\\ 1& -8& 18& -3\end{array}\right]$

Step-by-step solution

Step1:

a)

$A=\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]B=\left[\begin{array}{cc}0& 4\\ 1& 3\end{array}\right]$

$\begin{array}{r}AB=\left[\begin{array}{ll}0\cdot 2+1\cdot 1& 2\cdot 4+1\cdot 3\\ 3\cdot 0+2\cdot 1& 3\cdot 4+2\cdot 3\end{array}\right]\\ =\left[\begin{array}{ll}1& 11\\ 2& 18\end{array}\right]\end{array}$

Step 2:

b)

$\begin{array}{r}A=\left[\begin{array}{cc}1& -1\\ 0& 1\\ 2& 3\end{array}\right]B=\left[\begin{array}{ccc}3& -2& -1\\ 1& 0& 2\end{array}\right]\\ AB=\left[\begin{array}{ccc}1\cdot 3+(-1)\cdot 1& 1\cdot (-2)+(-1)\cdot 0& 1\cdot (-1)+(-1)\cdot 2\\ 0\cdot 3+1\cdot 1& 0\cdot (-2)+1\cdot 0& 0\cdot (-1)+1\cdot 2\\ 2\cdot 3+3\cdot 1& 2\cdot (-2)+3\cdot 0& 2\cdot (-1)+3\cdot 2\end{array}\right]\\ =\left[\begin{array}{ccc}2& -2& -3\\ 1& 0& 3\\ 9& -4& 4\end{array}\right]\end{array}$

Step3:

c)

$\begin{array}{r}A=\left[\begin{array}{cc}4& -3\\ 3& -1\\ 0& -2\\ -1& 5\end{array}\right]B=\left[\begin{array}{cccc}-1& 3& 2& -2\\ 0& -1& 4& -3\end{array}\right]\\ AB=\left[\begin{array}{cccc}4(-1)+(-3)0& 4\cdot 3+(-3)(-1)& 4\left(2\right)+(-3)4& 4(-2)+(-3)(-3)\\ 3(-1)+(-1)0& 3\left(3\right)+(-1)(-1)& 3\left(2\right)+(-1)4& 3(-2)+(-1)(-3)\\ 0(-1)+(-2)0& 0\left(3\right)+(-2)(-1)& 0\left(2\right)+(-2)4& 0(-2)+(-2)(-3)\\ -1(-1)+5\left(0\right)& -1\left(3\right)+5(-1)& -1\left(2\right)+5\left(4\right)& -1(-2)+5(-3)\end{array}\right]\\ =\left[\begin{array}{cccc}-4& 15& -4& 1\\ -3& 10& 2& -9\\ 0& 2& -8& 6\\ 1& -8& 18& -13\end{array}\right]\end{array}$