Question

Are the Matrices A and B inverses? Explain each step.

Answer 1

Given:

Given matrices are:

$A=\left[\begin{array}{ccc}1&0&2\\-2&1&1\\-1&1&2\end{array}\right]\ \text{and}\ B=\left[\begin{array}{ccc}-1&-2&2\\-3&-4&5\\1&1&-1\end{array}\right]$

To find:

We need to find whether the given matrices $A$ and $B$ are inverses or not.

Solution:

We know that, when we multiply a matrix with its inverse matrix, we get a Identity matrix.

$\Rightarrow AA^{-1}=I$

So, we will multiply the matrices $A$ and $B$ to check whether they are inverses or not.

$\Rightarrow AB=\left[ \begin{matrix} 1 & 0 & 2 \\ -2 & 1 & 1 \\ -1 & 1 & 2 \\\end{matrix} \right]\left[ \begin{matrix} -1 & -2 & 2 \\ -3 & -4 & 5 \\ 1 & 1 & -1 \\\end{matrix} \right]$

$\Rightarrow AB=\left[ \begin{matrix} -1+0+2 & -2+0+2 & 2+0-2 \\ 2-3+1 & 4-4+1 & -4+5-1 \\ 1-3+2 & 2-4+2 & -2+5-2 \\\end{matrix} \right]$

$\Rightarrow AB=\left[ \begin{matrix} -1+2 & -2+2 & 2-2 \\ 2+1-3 & 4+1-4 & -4-1+5 \\ 1+2-3 & 2+2-4 & -2-2+5 \\\end{matrix} \right]$

Rearranging the terms,

$\Rightarrow AB=\left[ \begin{matrix} -1+2 & -2+2 & 2-2 \\ 3-3 & 5-4 & -5+5 \\ 3-3 & 4-4 & -4+5 \\\end{matrix} \right]$

Simplifying the terms,

$\Rightarrow AB=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{matrix} \right]$

$\Rightarrow AB=I$

After multiplying, we got the answer as Identity matrix.

It means that, matrices $A$ and $B$ are inverses.

Therefore, given matrices $A$ and $B$ are inverses to each other.