# Proving that there is no invertible matrix with zero row sums using determinants

I have the following question which I know I should use the determinant to solve. Here it is:

Determine if there exists an invertible $3\times3$ matrix $A$ such that \begin{align*} a_{11}+a_{12}+a_{13}&=0 \\ a_{21}+a_{22}+a_{23}&=0 \\ a_{31}+a_{32}+a_{33}&=0 \end{align*}

I know that the answer is no, that there is no such matrix, but I’d like to know how to solve this question with determinants.

• You say you should use the determinant to solve this, but I would like to add that I think you should not do it this way. – JLA Aug 9 '14 at 21:08

The determinant is invariant under adding single multiples of one column to another. So using this we can add Column $2$ to Column $1$. Then add Column $3$ to Column $1$. This results in a column of zeroes. Hence, the determinant of $A$ is $0$ so it cannot be invertible.
• Perhaps I'm mistaken but you seem to impose a condition on $A$ that says if you add the columns together the resulting vector is the zero vector. – user71352 Aug 9 '14 at 20:48
{"ft_lang_label":"__label__en","ft_lang_prob":0.8856151,"math_prob":0.9799126,"size":402,"snap":"2019-35-2019-39","text_gpt3_token_len":135,"char_repetition_ratio":0.1281407,"word_repetition_ratio":0.0,"special_character_ratio":0.37064677,"punctuation_ratio":0.058139537,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9994172,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-22T23:14:11Z\",\"WARC-Record-ID\":\"<urn:uuid:d5ee9795-bcda-4f93-b84a-86688bcd1235>\",\"Content-Length\":\"139827\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:41c0191e-1927-4dab-b05d-2592b9d21131>\",\"WARC-Concurrent-To\":\"<urn:uuid:d3c95c25-8188-4c67-a610-db214149b80f>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/892386/proving-that-there-is-no-invertible-matrix-with-zero-row-sums-using-determinants\",\"WARC-Payload-Digest\":\"sha1:T5E6OSMCEZCOEPARHYMONTPGVJIUROR3\",\"WARC-Block-Digest\":\"sha1:4RJLPJOKBNWK6QOE57HBMNJNOWKDUD4E\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514575751.84_warc_CC-MAIN-20190922221623-20190923003623-00054.warc.gz\"}"}