The Multiplicative Inverse of a Matrix Show that B is the multiplicative inverse of A, where A = [-1 3 2 -5] and B = [5 3 2 1].

Question

The Multiplicative Inverse of a Matrix

Show that B is the multiplicative inverse of A, where

[latex]A=\left[\begin{array}{rr}-1 & 3 \ 2 & -5\end{array} ight][/latex] and [latex]B=\left[\begin{array}{ll}5 & 3 \ 2 & 1\end{array} ight][/latex].

Accepted Answer

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 [latex]I_2[/latex]; it is also a 2 × 2 matrix. We must show that

• [latex]A B=I_2=\left[\begin{array}{ll}1 & 0 \0 & 1\end{array} ight][/latex] and

• [latex]B A=I_2=\left[\begin{array}{ll}1 & 0 \0 & 1\end{array} ight][/latex].

Let's first show [latex]AB=I_2[/latex].

[latex]A B=\left[\begin{array}{rr}-1 & 3 \2 & -5\end{array} ight]\left[\begin{array}{ll}5 & 3 \2 & 1\end{array} ight][/latex]

[latex]=\left[\begin{array}{ll}-1(5)+3(2) & -1(3)+3(1) \2(5)+(-5)(2) & 2(3)+(-5)(1)\end{array} ight]=\left[\begin{array}{ll}1 & 0 \0 & 1\end{array} ight][/latex]

Let’s now show [latex]B_A = I_2.[/latex].

[latex]B A =\left[\begin{array}{ll}5 & 3 \2 & 1\end{array} ight]\left[\begin{array}{rr}-1 & 3 \2 & -5\end{array} ight][/latex]

[latex]=\left[\begin{array}{ll}5(-1)+3(2) & 5(3)+3(-5) \2(-1)+1(2) & 2(3)+1(-5)\end{array} ight]=\left[\begin{array}{ll}1 & 0 \0 & 1\end{array} ight][/latex]

Both products give the multiplicative identity matrix. Thus, B is the multiplicative inverse of A and we can designate B as [latex]A^{-1}=\left[\begin{array}{ll}5 & 3 \ 2 & 1\end{array} ight][/latex].

Question 8.4.1: The Multiplicative Inverse of a Matrix Show that B is the mu...

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