Why do the properties of determinants (used to calculate determinants from multiple matrices) apply not only...












3














A set of rules in my textbook is as follows:



a. If $A$ has a zero row (column), then $det A = 0$



b. If $B$ is obtained by interchanging two rows (columns) of $A$, then $det B = -det A$



c. If $A$ has two identical rows (columns), then $det A = 0$



d. If $B$ is obtained by multiplying a row (column) of $A$ by $k$, then $det B = kcdotdet A$



e. If $A$, $B$, and $C$ are identical except that the $i$-th row (column) of $C$ is the sum of the $i$-th rows (columns) of $A$ and $B$, then $det C = det B + det A$



f. If $B$ is obtained by adding a multiple of one row (column) of $A$ to another row (column), then $det B = det A$



I don't understand why "column" is in parentheses after every instance of row. Is that the same as saying, for example with clause a: "if $A$ has a zero row or a zero column, then $det A = 0$"? As in, it's saying that column and row can be used interchangably in the statement, as the statement holds true either way?



If the above interpretation is correct, then how? Why would these statements that apply to rows also apply to columns. The only situation I could see it applying is if the matrix is symmetrical, but the question doesn't specify that, it only says that the matrix is square.



Any help is appreciated.






linear-algebra matrices determinant







share|cite|improve this question









New contributor




Nest Doberman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • Yes, the same rules apply to columns too. This startled me as well when I was taught it. See J. G.'s answer. And welcome to the site.
    – timtfj
    15 mins ago
















3














A set of rules in my textbook is as follows:



a. If $A$ has a zero row (column), then $det A = 0$



b. If $B$ is obtained by interchanging two rows (columns) of $A$, then $det B = -det A$



c. If $A$ has two identical rows (columns), then $det A = 0$



d. If $B$ is obtained by multiplying a row (column) of $A$ by $k$, then $det B = kcdotdet A$



e. If $A$, $B$, and $C$ are identical except that the $i$-th row (column) of $C$ is the sum of the $i$-th rows (columns) of $A$ and $B$, then $det C = det B + det A$



f. If $B$ is obtained by adding a multiple of one row (column) of $A$ to another row (column), then $det B = det A$



I don't understand why "column" is in parentheses after every instance of row. Is that the same as saying, for example with clause a: "if $A$ has a zero row or a zero column, then $det A = 0$"? As in, it's saying that column and row can be used interchangably in the statement, as the statement holds true either way?



If the above interpretation is correct, then how? Why would these statements that apply to rows also apply to columns. The only situation I could see it applying is if the matrix is symmetrical, but the question doesn't specify that, it only says that the matrix is square.



Any help is appreciated.






linear-algebra matrices determinant







share|cite|improve this question









New contributor




Nest Doberman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • Yes, the same rules apply to columns too. This startled me as well when I was taught it. See J. G.'s answer. And welcome to the site.
    – timtfj
    15 mins ago














3












3








3


1





A set of rules in my textbook is as follows:



a. If $A$ has a zero row (column), then $det A = 0$



b. If $B$ is obtained by interchanging two rows (columns) of $A$, then $det B = -det A$



c. If $A$ has two identical rows (columns), then $det A = 0$



d. If $B$ is obtained by multiplying a row (column) of $A$ by $k$, then $det B = kcdotdet A$



e. If $A$, $B$, and $C$ are identical except that the $i$-th row (column) of $C$ is the sum of the $i$-th rows (columns) of $A$ and $B$, then $det C = det B + det A$



f. If $B$ is obtained by adding a multiple of one row (column) of $A$ to another row (column), then $det B = det A$



I don't understand why "column" is in parentheses after every instance of row. Is that the same as saying, for example with clause a: "if $A$ has a zero row or a zero column, then $det A = 0$"? As in, it's saying that column and row can be used interchangably in the statement, as the statement holds true either way?



If the above interpretation is correct, then how? Why would these statements that apply to rows also apply to columns. The only situation I could see it applying is if the matrix is symmetrical, but the question doesn't specify that, it only says that the matrix is square.



Any help is appreciated.






linear-algebra matrices determinant







share|cite|improve this question









New contributor




Nest Doberman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











A set of rules in my textbook is as follows:



a. If $A$ has a zero row (column), then $det A = 0$



b. If $B$ is obtained by interchanging two rows (columns) of $A$, then $det B = -det A$



c. If $A$ has two identical rows (columns), then $det A = 0$



d. If $B$ is obtained by multiplying a row (column) of $A$ by $k$, then $det B = kcdotdet A$



e. If $A$, $B$, and $C$ are identical except that the $i$-th row (column) of $C$ is the sum of the $i$-th rows (columns) of $A$ and $B$, then $det C = det B + det A$



f. If $B$ is obtained by adding a multiple of one row (column) of $A$ to another row (column), then $det B = det A$



I don't understand why "column" is in parentheses after every instance of row. Is that the same as saying, for example with clause a: "if $A$ has a zero row or a zero column, then $det A = 0$"? As in, it's saying that column and row can be used interchangably in the statement, as the statement holds true either way?



If the above interpretation is correct, then how? Why would these statements that apply to rows also apply to columns. The only situation I could see it applying is if the matrix is symmetrical, but the question doesn't specify that, it only says that the matrix is square.



Any help is appreciated.






linear-algebra matrices determinant




linear-algebra matrices determinant






share|cite|improve this question









New contributor




Nest Doberman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




Nest Doberman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 2 hours ago









amWhy

191k28224439




191k28224439






New contributor




Nest Doberman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 2 hours ago









Nest Doberman

161




161




New contributor




Nest Doberman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Nest Doberman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Nest Doberman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • Yes, the same rules apply to columns too. This startled me as well when I was taught it. See J. G.'s answer. And welcome to the site.
    – timtfj
    15 mins ago


















  • Yes, the same rules apply to columns too. This startled me as well when I was taught it. See J. G.'s answer. And welcome to the site.
    – timtfj
    15 mins ago
















Yes, the same rules apply to columns too. This startled me as well when I was taught it. See J. G.'s answer. And welcome to the site.
– timtfj
15 mins ago




Yes, the same rules apply to columns too. This startled me as well when I was taught it. See J. G.'s answer. And welcome to the site.
– timtfj
15 mins ago










1 Answer
1






active

oldest

votes


















6














The trick is that $det A^T=det A$. Therefore, if there's a zero column in $A$ there's a zero row in $A^T$, implying $det A^T=0$. You can check all the other copy-paste-to-column results in that way.






share|cite|improve this answer





















  • And, if you define determinants by the formula using signs of permutations, $det A^T=det A$ just amounts to the fact that the sign of a permutation is the same as the sign of its inverse.
    – Eric Wofsey
    1 hour ago










  • @EricWolsey Which, since signs multiply on composition of permutations, amounts to the primary-school fact that $(pm 1)^2=1$. (Or, if we work with parities, $1+1=2$ does it.)
    – J.G.
    1 hour ago













Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});






Nest Doberman is a new contributor. Be nice, and check out our Code of Conduct.










draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3058856%2fwhy-do-the-properties-of-determinants-used-to-calculate-determinants-from-multi%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









6














The trick is that $det A^T=det A$. Therefore, if there's a zero column in $A$ there's a zero row in $A^T$, implying $det A^T=0$. You can check all the other copy-paste-to-column results in that way.






share|cite|improve this answer





















  • And, if you define determinants by the formula using signs of permutations, $det A^T=det A$ just amounts to the fact that the sign of a permutation is the same as the sign of its inverse.
    – Eric Wofsey
    1 hour ago










  • @EricWolsey Which, since signs multiply on composition of permutations, amounts to the primary-school fact that $(pm 1)^2=1$. (Or, if we work with parities, $1+1=2$ does it.)
    – J.G.
    1 hour ago


















6














The trick is that $det A^T=det A$. Therefore, if there's a zero column in $A$ there's a zero row in $A^T$, implying $det A^T=0$. You can check all the other copy-paste-to-column results in that way.






share|cite|improve this answer





















  • And, if you define determinants by the formula using signs of permutations, $det A^T=det A$ just amounts to the fact that the sign of a permutation is the same as the sign of its inverse.
    – Eric Wofsey
    1 hour ago










  • @EricWolsey Which, since signs multiply on composition of permutations, amounts to the primary-school fact that $(pm 1)^2=1$. (Or, if we work with parities, $1+1=2$ does it.)
    – J.G.
    1 hour ago
















6












6








6






The trick is that $det A^T=det A$. Therefore, if there's a zero column in $A$ there's a zero row in $A^T$, implying $det A^T=0$. You can check all the other copy-paste-to-column results in that way.






share|cite|improve this answer












The trick is that $det A^T=det A$. Therefore, if there's a zero column in $A$ there's a zero row in $A^T$, implying $det A^T=0$. You can check all the other copy-paste-to-column results in that way.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 2 hours ago









J.G.

22.7k22136




22.7k22136












  • And, if you define determinants by the formula using signs of permutations, $det A^T=det A$ just amounts to the fact that the sign of a permutation is the same as the sign of its inverse.
    – Eric Wofsey
    1 hour ago










  • @EricWolsey Which, since signs multiply on composition of permutations, amounts to the primary-school fact that $(pm 1)^2=1$. (Or, if we work with parities, $1+1=2$ does it.)
    – J.G.
    1 hour ago




















  • And, if you define determinants by the formula using signs of permutations, $det A^T=det A$ just amounts to the fact that the sign of a permutation is the same as the sign of its inverse.
    – Eric Wofsey
    1 hour ago










  • @EricWolsey Which, since signs multiply on composition of permutations, amounts to the primary-school fact that $(pm 1)^2=1$. (Or, if we work with parities, $1+1=2$ does it.)
    – J.G.
    1 hour ago


















And, if you define determinants by the formula using signs of permutations, $det A^T=det A$ just amounts to the fact that the sign of a permutation is the same as the sign of its inverse.
– Eric Wofsey
1 hour ago




And, if you define determinants by the formula using signs of permutations, $det A^T=det A$ just amounts to the fact that the sign of a permutation is the same as the sign of its inverse.
– Eric Wofsey
1 hour ago












@EricWolsey Which, since signs multiply on composition of permutations, amounts to the primary-school fact that $(pm 1)^2=1$. (Or, if we work with parities, $1+1=2$ does it.)
– J.G.
1 hour ago






@EricWolsey Which, since signs multiply on composition of permutations, amounts to the primary-school fact that $(pm 1)^2=1$. (Or, if we work with parities, $1+1=2$ does it.)
– J.G.
1 hour ago












Nest Doberman is a new contributor. Be nice, and check out our Code of Conduct.










draft saved

draft discarded


















Nest Doberman is a new contributor. Be nice, and check out our Code of Conduct.













Nest Doberman is a new contributor. Be nice, and check out our Code of Conduct.












Nest Doberman is a new contributor. Be nice, and check out our Code of Conduct.
















Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3058856%2fwhy-do-the-properties-of-determinants-used-to-calculate-determinants-from-multi%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Eastern Orthodox Church

Image
Not to be confused with Oriental Orthodoxy, Eastern Christianity, or Eastern Catholic Churches. "Eastern Orthodoxy", "Orthodox Church", "Orthodox Catholic Church", and "Orthodox Christian Church" redirect here. For other uses, see Orthodox (disambiguation). This article may be too long to read and navigate comfortably . The readable prose size is 132 kilobytes. Please consider splitting content into sub-articles, condensing it, or adding or removing subheadings. ( January 2018 ) Part of a series on the Eastern Orthodox Church Mosaic of Christ Pantocrator, Hagia Sophia Overview Structure Theology (History of theology) Liturgy Church history Holy Mysteries View of salvation View of Mary View of icons Background Crucifixion / Resurrection / Ascension of Jesus Christianity Christian Church Apostolic succession Four Marks of the Church Orthodoxy Organization Autocephaly Patriarchate ...
Read more

Zagreb

Image
This article is about the Croatian capital city. For other uses, see Zagreb (disambiguation). City in City of Zagreb, Croatia Zagreb City Grad Zagreb City of Zagreb Clockwise, from top: St. Mark's Square, Croatian State Archives, Zagreb Tram, Art Pavilion, Cibona & HOTO towers and Croatian National Theatre. Flag Coat of arms Zagreb Location of Zagreb in Croatia Show map of Croatia Zagreb Zagreb (Europe) Show map of Europe Coordinates: 45°49′N 15°59′E  /  45.817°N 15.983°E  / 45.817; 15.983 Coordinates: 45°49′N 15°59′E  /  45.817°N 15.983°E  / 45.817; 15.983 Country   Croatia County City of Zagreb RC diocese 1094 Free royal city 1242 Unified 1850 Subdivisions 17 city districts 218 local committees (70 settlements) Government  • Type Mayor-Council  • Mayor Milan Bandić (BM 365)  • City Assembly 51 members BM 365, ZL, NS-R (14) SDP, GLAS, HNS, HSS, NH-PS (13) ...
Read more

Understanding the information contained in the Deep Space Network XML data?

Image
up vote 1 down vote favorite NASA's Deep Space Network activity can be viewed conveniently at https://eyes.nasa.gov/dsn/dsn.html but if you would like data in a more raw XML form, it can be accessed as described in this answer Below is a small Python script that reads the XML into a list of dictionaries, and saves it to disk as a JSON. The dictionaries are organized at the higher level by dish, then by spacecraft per dish. Question: What are the most-likely meanings of all the data field contained in the Deep Space Network XML data? This is the example url from the other answer, used in the script below: https://eyes.nasa.gov/dsn/data/1365107113.xml def dictify(r,root=True): # https://stackoverflow.com/questions/2148119/how-to-convert-an-xml-string-to-a-dictionary-in-python/30923963#30923963 if root: ret...
Read more