Question

Show that$\left[\begin{array}{cc}2& 3\\ 1& 2\\ -1& -1\end{array}\begin{array}{c}-1\\ 1\\ 3\end{array}\right]$is the inverse ofrole="math" localid="1668443022252" $\left[\begin{array}{ccc}7& 8& 5\\ 4& 5& -3\\ 1& -1& 1\end{array}\right]$ .

Short answer

Since, $\left[\begin{array}{ccc}2& 3& -1\\ 1& 2& 1\\ -1& -1& 3\end{array}\right]\left[\begin{array}{ccc}7& -8& 5\\ -4& 5& -3\\ 1& -1& 1\end{array}\right]=\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right]\text{, so}\left[\begin{array}{ccc}7& -8& 5\\ -4& 5& -3\\ 1& -1& 1\end{array}\right]$ is the inverse of$\left[\begin{array}{ccc}2& 3& -1\\ 1& 2& 1\\ -1& -1& 3\end{array}\right]$ .

Step-by-step solution

Step 1

Multiply the two matrices together. If their product equals the identity matrix, then they are inverses of each other.

$\begin{array}{r}\left[\begin{array}{ccc}2& 3& -1\\ 1& 2& 1\\ -1& -1& 3\end{array}\right]\left[\begin{array}{ccc}7& -8& 5\\ -4& 5& -3\\ 1& -1& 1\end{array}\right]\\ =\left[\begin{array}{lll}2\left(7\right)+3(-4)+(-1)\left(1\right)& 2(-8)+3\left(5\right)+(-1)(-1)& 2\left(5\right)+3(-3)+(-1)\left(1\right)\\ 1\left(7\right)+\left(2\right)(-4)+1\left(1\right)& 1(-8)+2\left(5\right)+1(-1)& 1\left(5\right)+2(-3)+1\left(1\right)\\ -1\left(7\right)+(-1)(-4)+3\left(1\right)& (-1)(-8)+(-1)\left(5\right)+3(-1)& (-1)\left(5\right)+(-1)(-3)+3\left(1\right)\end{array}\right]\\ =\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\end{array}$