Question 8.4.1: The Multiplicative Inverse of a Matrix Show that B is the mu...
The Multiplicative Inverse of a Matrix
Show that B is the multiplicative inverse of A, where
A=\left[\begin{array}{rr}-1 & 3 \\ 2 & -5\end{array}\right] and B=\left[\begin{array}{ll}5 & 3 \\ 2 & 1\end{array}\right].
To show that B is the multiplicative inverse of A, we must find the products AB and BA. If B is the multiplicative inverse of A, then AB will be the multiplicative identity matrix and BA will be the multiplicative identity matrix. Because A and B are 2 × 2 matrices, n = 2. Thus, we denote the multiplicative identity matrix as I_2; it is also a 2 × 2 matrix. We must show that
• A B=I_2=\left[\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right] and
• B A=I_2=\left[\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right].
Let’s first show AB=I_2.
A B=\left[\begin{array}{rr}-1 & 3 \\2 & -5\end{array}\right]\left[\begin{array}{ll}5 & 3 \\2 & 1\end{array}\right]
=\left[\begin{array}{ll}-1(5)+3(2) & -1(3)+3(1) \\2(5)+(-5)(2) & 2(3)+(-5)(1)\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right]
Let’s now show B_A = I_2..
B A =\left[\begin{array}{ll}5 & 3 \\2 & 1\end{array}\right]\left[\begin{array}{rr}-1 & 3 \\2 & -5\end{array}\right]=\left[\begin{array}{ll}5(-1)+3(2) & 5(3)+3(-5) \\2(-1)+1(2) & 2(3)+1(-5)\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right]
Both products give the multiplicative identity matrix. Thus, B is the multiplicative inverse of A and we can designate B as A^{-1}=\left[\begin{array}{ll}5 & 3 \\ 2 & 1\end{array}\right].