Question

Find AB, if \[A=\left[ \begin{matrix}
   1 & 2 & 3 \\
   1 & -2 & 3 \\
\end{matrix} \right]\] and \[B=\left[ \begin{matrix}
   1 & -1 \\
   1 & 2 \\
   1 & -2 \\
\end{matrix} \right]\]. Examine whether AB has inverse or not.

Answer 1

Hint: We first need to find the multiplication value of two matrices. We check if the matrix multiplication is valid or not. Then we explain the condition of a matrix is invertible. We find the determinant value of the matrix AB. The matrix being non-singular we can say that the matrix has an inverse.

Complete step by step answer:
First, we need matrix multiplication of two given matrices \[A=\left[ \begin{matrix}
   1 & 2 & 3 \\
   1 & -2 & 3 \\
\end{matrix} \right]\] and \[B=\left[ \begin{matrix}
   1 & -1 \\
   1 & 2 \\
   1 & -2 \\
\end{matrix} \right]\].
For matrix multiplication, we need to have same value for the column of the first matrix and the row of the second matrix. In this case of AB, we have multiplication of a $ \left( 2\times 3 \right) $ matrix to a $ \left( 3\times 2 \right) $ matrix which is possible.
So, we have to find the multiplied matrix of AB. So, \[AB=\left[ \begin{matrix}
   1 & 2 & 3 \\
   1 & -2 & 3 \\
\end{matrix} \right]\left[ \begin{matrix}
   1 & -1 \\
   1 & 2 \\
   1 & -2 \\
\end{matrix} \right]\].
We expand the multiplication as \[AB=\left[ \begin{matrix}
   1+2+3 & -1+4-6 \\
   1-2+3 & -1-4-6 \\
\end{matrix} \right]=\left[ \begin{matrix}
   6 & -3 \\
   2 & -11 \\
\end{matrix} \right]\].
Now we need to find whether AB has inverse or not.
If the given matrix is singular which means if the determinant value of the matrix is 0 then the matrix doesn’t have an inverse. If the matrix is non-singular then the matrix has an inverse.
So, to find if AB has an inverse or not, we need to find the value of $ \det \left( AB \right) $ .
We need to expand the matrix \[AB=\left[ \begin{matrix}
   6 & -3 \\
   2 & -11 \\
\end{matrix} \right]\] along the first row.
We have $ \det \left( AB \right)=6\times \left( -11 \right)-2\times \left( -3 \right)=-66+6=-60 $ .
So, the determinant value of the matrix AB is not 0 which means the matrix is invertible.
Therefore, AB is invertible.

Note:
 We can’t use the actual matrices A and B separately to find if AB has an inverse or not. There is no direct relation to find the determinant value of AB from the matrices A and B. Multiplication of the matrices is necessary to find the determinant value of the matrix AB.