031 Review Part 1, Problem 9

True or false: If  $A$  is an invertible  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3\times 3}   matrix, and  $B$  and  $C$  are  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3\times 3}   matrices such that  $AB=AC,$ 

then  $B=C.$

Solution:
Since  $A$  is invertible,  $A^{-1}$  exists.
Since  $AB=AC,$  we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A^{-1}(AB)=A^{-1}(AC).}
Then, by associativity of matrix multiplication, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_3}   is the  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3}   identity matrix.
Hence, the statement is true.

Final Answer:
TRUE

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