Question

Find the product AB, where

a)$\mathit{A}\mathbf{=}\left[\begin{array}{ccc}1& 0& 1\\ 0& -1& -1\\ -1& 1& 0\end{array}\right]\mathit{B}\mathbf{=}\left[\begin{array}{ccc}0& 1& -1\\ 1& -1& 0\\ -1& 0& 1\end{array}\right]$

b)role="math" localid="1668435813041" $\mathit{A}\mathbf{=}\left[\begin{array}{ccc}1& -3& 0\\ 1& 2& 2\\ 2& 1& 4\end{array}\right]\mathbf{=}\left[\begin{array}{ccccc}1& -1& 2& 3& \\ 1& 0& 3& -1& \\ 3& -2& 0& 2& \end{array}\right]$

c)$\mathit{A}\mathbf{=}\left[\begin{array}{cc}0& -1\\ 7& 2\\ -4& -3\end{array}\right]\mathit{B}\mathbf{=}\left[\begin{array}{cccccc}4& -1& 2& 3& 0& \\ -2& 0& 3& 4& 1& \end{array}\right]$

Short answer

a) Required answer is$AB=\left[\begin{array}{ccc}-1& 1& 0\\ 0& 1& -1\\ 1& -2& 1\end{array}\right]$

b) Required answer is$AB=\left[\begin{array}{cccc}4& -1& -7& 6\\ -7& -5& 8& 5\\ 4& 0& 7& 3\end{array}\right]$

c) Required answer is$AB=\left[\begin{array}{ccccc}2& 0& -3& -4& -1\\ 24& -7& 20& 29& 2\\ 10& 4& -17& -44& -3\end{array}\right]$

Step-by-step solution

Multiplication of matrices

a)

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

Multiplication of matrices

b)

$\begin{array}{r}A=\left[\begin{array}{ccc}1& -3& 0\\ 1& 2& 2\\ 2& 1& -1\end{array}\right]B=\left[\begin{array}{cccc}1& -1& 2& 3\\ -1& 0& 3& -1\\ -3& -2& 0& 2\end{array}\right]\\ AB=\left[\begin{array}{cccc}1\left(1\right)+(-3)\cdot (-1)+0(-3)& 1(-1)+(-3)0+0(-2)& 1\left(2\right)+(-3)3+0\cdot 0& 1\left(3\right)+(-3)(-1)+0\cdot 2\\ 1\cdot 1+2(-1)+2(-3)& 1(-1)+\left(2\right)0+2(-2)& 1\cdot 2+2\cdot 3+2\cdot 0& 1\left(3\right)+2(-1)+2\cdot 2\\ 2\cdot 1+1(-1)+(-1)(-3)& 2(-1)+1\cdot 0+(-1)(-2)& 2\cdot 2+1\cdot 3+(-1)0& 2\cdot 3+1(-1)+(-1)2\end{array}\right]\\ AB=\left[\begin{array}{cccc}4& -1& -7& 6\\ -7& -5& 8& 5\\ 4& 0& 7& 3\end{array}\right]\end{array}$

Multiplication of matrices

c)

$\begin{array}{r}A=\left[\begin{array}{cc}0& -1\\ 7& 2\\ -4& -3\end{array}\right]B=\left[\begin{array}{ccccc}4& -1& 2& 3& 0\\ -2& 0& 3& 4& 1\end{array}\right]\\ AB=\left[\begin{array}{ccccc}0\cdot 4+(-1)\cdot (-2)& 0\cdot (-1)+(-1)\cdot 0& 0\cdot \left(2\right)+(-1)\cdot 3& 0\cdot 3+(-1)\cdot 4& 0\cdot 0+(-1)\cdot 1\\ 7\cdot 4+2\cdot (-2)& 7\cdot (-1)+2\cdot 0& 7\cdot 2+2\cdot 3& 7\cdot \left(3\right)+2\cdot \left(4\right)& 7\cdot \left(0\right)+2\cdot \left(1\right)\\ -4\cdot \left(4\right)+(-3)\cdot (-2)& -4\cdot (-1)+(-3)\cdot 0& -4\cdot \left(2\right)+(-3)\cdot 3& -4\cdot \left(3\right)+(-3)\cdot 4& -4\cdot \left(0\right)+(-3)\cdot 1\end{array}\right]\\ AB=\left[\begin{array}{ccccc}2& 0& -3& -4& -1\\ 24& -7& 20& 29& 2\\ 10& 4& -17& -24& -3\end{array}\right]\end{array}$