Question

Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix In.) [01000−21−10]and[1011000−10]\left[ \begin{matrix} 
0 & 1 & 0 \
0 & 0 & -2 \
1 & -1 & 0 
\end{matrix} ight]
\quad 	ext{and} \quad
\left[ \begin{matrix} 
1 & 0 & 1 \
1 & 0 & 0 \
0 & -1 & 0 
\end{matrix} ight]

Accepted Answer

Recall that two matrices $$ A $$ and $$ B $$ are inverses of each other if and only if their product in both orders equals the identity matrix $$ I_n $$, where $$ n  is $$the size of the matrices. For 3x3 matrices, $$ I_3  is $$the 3x3 identity matrix. Calculate the product $$ A 	imes B  by $$multiplying the rows of matrix $$ A  by $$the columns of matrix $$ B . $$Each element $$ c_{ij}  of $$the product matrix is found by $$ c_{ij} = \sum_{k=1}^3 a_{ik} b_{kj} . $$Calculate the product $$ B 	imes A $$ similarly, by multiplying the rows of matrix $$ B  by $$the columns of matrix $$ A . $$Compare both products $$ A 	imes B $$ and $$ B 	imes A  to $$the 3x3 identity matrix $$ I_3 $$, which has 1's on the diagonal and 0's elsewhere. If both products equal $$ I_3 $$, then $$ A $$ and $$ B $$ are inverses of each other. If either product does not equal $$ I_3 $$, then they are not inverses.

Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix I_n.)
$[\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & - 2 \\ 1 & - 1 & 0 \end{matrix}]$

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Key Concepts

Matrix Multiplication

Identity Matrix

Matrix Inverse