Find the products AB and BA to determine whether B is the multiplicative inverse of A. ? a. B ≠ A^-1 b. B = A-1

Apr 21, 2018

B is the multiplicative inverse of A and A is the multiplicative inverse of B.

Explanation:

Let's multiply AB first.

First Row, First Column
$5 \cdot 2 + 3 \cdot \left(- 3\right) = 1$

First Row, Second Column
$5 \left(- 3\right) + 3 \cdot 5 = 0$

Second Row, First Column
$3 \cdot 2 + 2 \left(- 3\right) = 0$

Second Row, Second Column
$3 \left(- 3\right) + 2 \cdot 5 = 1$

So AB = $\frac{1 , 0}{0 , 1}$ which is the identity matrix

Lets try BA now.

First Row, First Column
$2 \cdot 5 + 3 \cdot \left(- 3\right) = 1$

First Row, Second Column
$3 \cdot 2 + 2 \left(- 3\right) = 0$

Second Row, First Column
$\left(- 5\right) \cdot 3 + 5 \cdot 3 = 0$

Second Row, Second Column
$3 \left(- 3\right) + 2 \cdot 5 = 1$

So BA = $\frac{1 , 0}{0 , 1}$ which is also the identity matrix

A and B are multiplicative inverses.