031 Review Part 1, Problem 9

True or false: If  $A$  is an invertible  $3\times 3$  matrix, and  $B$  and  $C$  are  $3\times 3$  matrices such that  $AB=AC,$ 

then  $B=C.$

Solution:
Since  $A$  is invertible,  $A^{-1}$  exists.
Since  $AB=AC,$  we have
$A^{-1}(AB)=A^{-1}(AC).$
Then, by associativity of matrix multiplication, we have
${\begin{array}{rcl}\displaystyle {(A^{-1}A)B}&=&\displaystyle {(A^{-1}A)C}\\&&\\\displaystyle {I_{3}B}&=&\displaystyle {I_{3}C}\\&&\\\displaystyle {B}&=&\displaystyle {C}\end{array}}$
where  $I_{3}$  is the  $3\times 3$  identity matrix.
Hence, the statement is true.

Final Answer:
TRUE

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