$AC^T = det(A)I$












1














Let A = $begin{bmatrix} a_{11} & a_{12} & dots & a_{1n} \ a_{21} & a_{22} & dots & a_{2n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} &
dots & a_{nn} end{bmatrix}$

and the matrix of cofactors of $A$ is
$$C=begin{bmatrix} C_{11} & C_{12} & dots & C_{1n} \ C_{21} & C_{22} & dots & C_{2n} \ vdots & vdots & ddots & vdots \ C_{n1} & C_{n2} &
dots & C_{nn} end{bmatrix}.
$$

I try to understand why $AC^T = det(A)I$ necessarily.



Why is it that $a_{11}C_{21} + a_{12}C_{22} + dots + a_{1n}C_{2n} = 0$?










share|cite|improve this question





























    1














    Let A = $begin{bmatrix} a_{11} & a_{12} & dots & a_{1n} \ a_{21} & a_{22} & dots & a_{2n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} &
    dots & a_{nn} end{bmatrix}$

    and the matrix of cofactors of $A$ is
    $$C=begin{bmatrix} C_{11} & C_{12} & dots & C_{1n} \ C_{21} & C_{22} & dots & C_{2n} \ vdots & vdots & ddots & vdots \ C_{n1} & C_{n2} &
    dots & C_{nn} end{bmatrix}.
    $$

    I try to understand why $AC^T = det(A)I$ necessarily.



    Why is it that $a_{11}C_{21} + a_{12}C_{22} + dots + a_{1n}C_{2n} = 0$?










    share|cite|improve this question



























      1












      1








      1







      Let A = $begin{bmatrix} a_{11} & a_{12} & dots & a_{1n} \ a_{21} & a_{22} & dots & a_{2n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} &
      dots & a_{nn} end{bmatrix}$

      and the matrix of cofactors of $A$ is
      $$C=begin{bmatrix} C_{11} & C_{12} & dots & C_{1n} \ C_{21} & C_{22} & dots & C_{2n} \ vdots & vdots & ddots & vdots \ C_{n1} & C_{n2} &
      dots & C_{nn} end{bmatrix}.
      $$

      I try to understand why $AC^T = det(A)I$ necessarily.



      Why is it that $a_{11}C_{21} + a_{12}C_{22} + dots + a_{1n}C_{2n} = 0$?










      share|cite|improve this question















      Let A = $begin{bmatrix} a_{11} & a_{12} & dots & a_{1n} \ a_{21} & a_{22} & dots & a_{2n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} &
      dots & a_{nn} end{bmatrix}$

      and the matrix of cofactors of $A$ is
      $$C=begin{bmatrix} C_{11} & C_{12} & dots & C_{1n} \ C_{21} & C_{22} & dots & C_{2n} \ vdots & vdots & ddots & vdots \ C_{n1} & C_{n2} &
      dots & C_{nn} end{bmatrix}.
      $$

      I try to understand why $AC^T = det(A)I$ necessarily.



      Why is it that $a_{11}C_{21} + a_{12}C_{22} + dots + a_{1n}C_{2n} = 0$?







      linear-algebra determinant laplace-expansion






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 4 at 15:04









      A.Γ.

      22.6k32656




      22.6k32656










      asked Jan 4 at 14:24









      Kid CudiKid Cudi

      456




      456






















          1 Answer
          1






          active

          oldest

          votes


















          1














          The expression
          $$
          color{red}{a_{11}}C_{21} + color{red}{a_{12}}C_{22} + dots + color{red}{a_{1n}}C_{2n}
          $$

          is the Laplace expansion along the second row of
          $$
          begin{vmatrix}
          a_{11} & a_{12} & dots & a_{1n} \ color{red}{a_{11}} & color{red}{a_{12}} & color{red}dots & color{red}{a_{1n}} \ a_{31} & a_{32} & dots & a_{3n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} &
          dots & a_{nn}
          end{vmatrix}=color{red}{a_{11}}C_{21} + color{red}{a_{12}}C_{22} + dots + color{red}{a_{1n}}C_{2n}=0.
          $$



          Edit: take the matrix $A$ and do the determinant expansion along the second row
          $$
          begin{vmatrix}
          a_{11} & a_{12} & dots & a_{1n} \ color{blue}{a_{21}} & color{blue}{a_{22}} & color{blue}dots & color{blue}{a_{2n}} \ a_{31} & a_{32} & dots & a_{3n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} & dots & a_{nn}
          end{vmatrix}=color{blue}{a_{21}}C_{21} + color{blue}{a_{22}}C_{22} + dots + color{blue}{a_{2n}}C_{2n}.
          $$

          Observe that the blue elements are not used to build $C_{2j}$. If we replace the blue elements with the red elements above we get exactly what we need.






          share|cite|improve this answer























          • Where did that determinant come from? Why do the first two rows suddenly become the same? Sorry I'm slow, don't see the jump. I got the same answer from the book.
            – Kid Cudi
            Jan 4 at 14:54








          • 1




            @KidCudi Yes, it may take time to understand. The determinant expansion of a matrix is the sum of the element${}_{ij}cdot$the cofactor${}_{ij}$. The expression has the same structure as the determinant expansion along the second row. The red elements must be the elements of the row and the cofactors are the same as for the matrix $A$ for the same row. Observe that the second row is not used when building the cofactors. Try to do the expanstion for this matrix yourself.
            – A.Γ.
            Jan 4 at 15:00








          • 1




            @KidCudi I have edited the answer, hopefully it is clearer now.
            – A.Γ.
            Jan 4 at 15:13






          • 1




            @KidCudi I am saying that the expression in question $a_{11}C_{21}+ldots$ is the determinant expansion of $B$, which is zero. Then the expression is also zero.
            – A.Γ.
            Jan 4 at 15:17






          • 1




            I have added a $3$-rd row to your two matrices to make the pattern behind the vdots clearer.
            – darij grinberg
            Jan 4 at 20:26











          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
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3061697%2fact-detai%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









          1














          The expression
          $$
          color{red}{a_{11}}C_{21} + color{red}{a_{12}}C_{22} + dots + color{red}{a_{1n}}C_{2n}
          $$

          is the Laplace expansion along the second row of
          $$
          begin{vmatrix}
          a_{11} & a_{12} & dots & a_{1n} \ color{red}{a_{11}} & color{red}{a_{12}} & color{red}dots & color{red}{a_{1n}} \ a_{31} & a_{32} & dots & a_{3n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} &
          dots & a_{nn}
          end{vmatrix}=color{red}{a_{11}}C_{21} + color{red}{a_{12}}C_{22} + dots + color{red}{a_{1n}}C_{2n}=0.
          $$



          Edit: take the matrix $A$ and do the determinant expansion along the second row
          $$
          begin{vmatrix}
          a_{11} & a_{12} & dots & a_{1n} \ color{blue}{a_{21}} & color{blue}{a_{22}} & color{blue}dots & color{blue}{a_{2n}} \ a_{31} & a_{32} & dots & a_{3n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} & dots & a_{nn}
          end{vmatrix}=color{blue}{a_{21}}C_{21} + color{blue}{a_{22}}C_{22} + dots + color{blue}{a_{2n}}C_{2n}.
          $$

          Observe that the blue elements are not used to build $C_{2j}$. If we replace the blue elements with the red elements above we get exactly what we need.






          share|cite|improve this answer























          • Where did that determinant come from? Why do the first two rows suddenly become the same? Sorry I'm slow, don't see the jump. I got the same answer from the book.
            – Kid Cudi
            Jan 4 at 14:54








          • 1




            @KidCudi Yes, it may take time to understand. The determinant expansion of a matrix is the sum of the element${}_{ij}cdot$the cofactor${}_{ij}$. The expression has the same structure as the determinant expansion along the second row. The red elements must be the elements of the row and the cofactors are the same as for the matrix $A$ for the same row. Observe that the second row is not used when building the cofactors. Try to do the expanstion for this matrix yourself.
            – A.Γ.
            Jan 4 at 15:00








          • 1




            @KidCudi I have edited the answer, hopefully it is clearer now.
            – A.Γ.
            Jan 4 at 15:13






          • 1




            @KidCudi I am saying that the expression in question $a_{11}C_{21}+ldots$ is the determinant expansion of $B$, which is zero. Then the expression is also zero.
            – A.Γ.
            Jan 4 at 15:17






          • 1




            I have added a $3$-rd row to your two matrices to make the pattern behind the vdots clearer.
            – darij grinberg
            Jan 4 at 20:26
















          1














          The expression
          $$
          color{red}{a_{11}}C_{21} + color{red}{a_{12}}C_{22} + dots + color{red}{a_{1n}}C_{2n}
          $$

          is the Laplace expansion along the second row of
          $$
          begin{vmatrix}
          a_{11} & a_{12} & dots & a_{1n} \ color{red}{a_{11}} & color{red}{a_{12}} & color{red}dots & color{red}{a_{1n}} \ a_{31} & a_{32} & dots & a_{3n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} &
          dots & a_{nn}
          end{vmatrix}=color{red}{a_{11}}C_{21} + color{red}{a_{12}}C_{22} + dots + color{red}{a_{1n}}C_{2n}=0.
          $$



          Edit: take the matrix $A$ and do the determinant expansion along the second row
          $$
          begin{vmatrix}
          a_{11} & a_{12} & dots & a_{1n} \ color{blue}{a_{21}} & color{blue}{a_{22}} & color{blue}dots & color{blue}{a_{2n}} \ a_{31} & a_{32} & dots & a_{3n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} & dots & a_{nn}
          end{vmatrix}=color{blue}{a_{21}}C_{21} + color{blue}{a_{22}}C_{22} + dots + color{blue}{a_{2n}}C_{2n}.
          $$

          Observe that the blue elements are not used to build $C_{2j}$. If we replace the blue elements with the red elements above we get exactly what we need.






          share|cite|improve this answer























          • Where did that determinant come from? Why do the first two rows suddenly become the same? Sorry I'm slow, don't see the jump. I got the same answer from the book.
            – Kid Cudi
            Jan 4 at 14:54








          • 1




            @KidCudi Yes, it may take time to understand. The determinant expansion of a matrix is the sum of the element${}_{ij}cdot$the cofactor${}_{ij}$. The expression has the same structure as the determinant expansion along the second row. The red elements must be the elements of the row and the cofactors are the same as for the matrix $A$ for the same row. Observe that the second row is not used when building the cofactors. Try to do the expanstion for this matrix yourself.
            – A.Γ.
            Jan 4 at 15:00








          • 1




            @KidCudi I have edited the answer, hopefully it is clearer now.
            – A.Γ.
            Jan 4 at 15:13






          • 1




            @KidCudi I am saying that the expression in question $a_{11}C_{21}+ldots$ is the determinant expansion of $B$, which is zero. Then the expression is also zero.
            – A.Γ.
            Jan 4 at 15:17






          • 1




            I have added a $3$-rd row to your two matrices to make the pattern behind the vdots clearer.
            – darij grinberg
            Jan 4 at 20:26














          1












          1








          1






          The expression
          $$
          color{red}{a_{11}}C_{21} + color{red}{a_{12}}C_{22} + dots + color{red}{a_{1n}}C_{2n}
          $$

          is the Laplace expansion along the second row of
          $$
          begin{vmatrix}
          a_{11} & a_{12} & dots & a_{1n} \ color{red}{a_{11}} & color{red}{a_{12}} & color{red}dots & color{red}{a_{1n}} \ a_{31} & a_{32} & dots & a_{3n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} &
          dots & a_{nn}
          end{vmatrix}=color{red}{a_{11}}C_{21} + color{red}{a_{12}}C_{22} + dots + color{red}{a_{1n}}C_{2n}=0.
          $$



          Edit: take the matrix $A$ and do the determinant expansion along the second row
          $$
          begin{vmatrix}
          a_{11} & a_{12} & dots & a_{1n} \ color{blue}{a_{21}} & color{blue}{a_{22}} & color{blue}dots & color{blue}{a_{2n}} \ a_{31} & a_{32} & dots & a_{3n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} & dots & a_{nn}
          end{vmatrix}=color{blue}{a_{21}}C_{21} + color{blue}{a_{22}}C_{22} + dots + color{blue}{a_{2n}}C_{2n}.
          $$

          Observe that the blue elements are not used to build $C_{2j}$. If we replace the blue elements with the red elements above we get exactly what we need.






          share|cite|improve this answer














          The expression
          $$
          color{red}{a_{11}}C_{21} + color{red}{a_{12}}C_{22} + dots + color{red}{a_{1n}}C_{2n}
          $$

          is the Laplace expansion along the second row of
          $$
          begin{vmatrix}
          a_{11} & a_{12} & dots & a_{1n} \ color{red}{a_{11}} & color{red}{a_{12}} & color{red}dots & color{red}{a_{1n}} \ a_{31} & a_{32} & dots & a_{3n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} &
          dots & a_{nn}
          end{vmatrix}=color{red}{a_{11}}C_{21} + color{red}{a_{12}}C_{22} + dots + color{red}{a_{1n}}C_{2n}=0.
          $$



          Edit: take the matrix $A$ and do the determinant expansion along the second row
          $$
          begin{vmatrix}
          a_{11} & a_{12} & dots & a_{1n} \ color{blue}{a_{21}} & color{blue}{a_{22}} & color{blue}dots & color{blue}{a_{2n}} \ a_{31} & a_{32} & dots & a_{3n} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} & dots & a_{nn}
          end{vmatrix}=color{blue}{a_{21}}C_{21} + color{blue}{a_{22}}C_{22} + dots + color{blue}{a_{2n}}C_{2n}.
          $$

          Observe that the blue elements are not used to build $C_{2j}$. If we replace the blue elements with the red elements above we get exactly what we need.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 4 at 20:26









          darij grinberg

          10.2k33062




          10.2k33062










          answered Jan 4 at 14:31









          A.Γ.A.Γ.

          22.6k32656




          22.6k32656












          • Where did that determinant come from? Why do the first two rows suddenly become the same? Sorry I'm slow, don't see the jump. I got the same answer from the book.
            – Kid Cudi
            Jan 4 at 14:54








          • 1




            @KidCudi Yes, it may take time to understand. The determinant expansion of a matrix is the sum of the element${}_{ij}cdot$the cofactor${}_{ij}$. The expression has the same structure as the determinant expansion along the second row. The red elements must be the elements of the row and the cofactors are the same as for the matrix $A$ for the same row. Observe that the second row is not used when building the cofactors. Try to do the expanstion for this matrix yourself.
            – A.Γ.
            Jan 4 at 15:00








          • 1




            @KidCudi I have edited the answer, hopefully it is clearer now.
            – A.Γ.
            Jan 4 at 15:13






          • 1




            @KidCudi I am saying that the expression in question $a_{11}C_{21}+ldots$ is the determinant expansion of $B$, which is zero. Then the expression is also zero.
            – A.Γ.
            Jan 4 at 15:17






          • 1




            I have added a $3$-rd row to your two matrices to make the pattern behind the vdots clearer.
            – darij grinberg
            Jan 4 at 20:26


















          • Where did that determinant come from? Why do the first two rows suddenly become the same? Sorry I'm slow, don't see the jump. I got the same answer from the book.
            – Kid Cudi
            Jan 4 at 14:54








          • 1




            @KidCudi Yes, it may take time to understand. The determinant expansion of a matrix is the sum of the element${}_{ij}cdot$the cofactor${}_{ij}$. The expression has the same structure as the determinant expansion along the second row. The red elements must be the elements of the row and the cofactors are the same as for the matrix $A$ for the same row. Observe that the second row is not used when building the cofactors. Try to do the expanstion for this matrix yourself.
            – A.Γ.
            Jan 4 at 15:00








          • 1




            @KidCudi I have edited the answer, hopefully it is clearer now.
            – A.Γ.
            Jan 4 at 15:13






          • 1




            @KidCudi I am saying that the expression in question $a_{11}C_{21}+ldots$ is the determinant expansion of $B$, which is zero. Then the expression is also zero.
            – A.Γ.
            Jan 4 at 15:17






          • 1




            I have added a $3$-rd row to your two matrices to make the pattern behind the vdots clearer.
            – darij grinberg
            Jan 4 at 20:26
















          Where did that determinant come from? Why do the first two rows suddenly become the same? Sorry I'm slow, don't see the jump. I got the same answer from the book.
          – Kid Cudi
          Jan 4 at 14:54






          Where did that determinant come from? Why do the first two rows suddenly become the same? Sorry I'm slow, don't see the jump. I got the same answer from the book.
          – Kid Cudi
          Jan 4 at 14:54






          1




          1




          @KidCudi Yes, it may take time to understand. The determinant expansion of a matrix is the sum of the element${}_{ij}cdot$the cofactor${}_{ij}$. The expression has the same structure as the determinant expansion along the second row. The red elements must be the elements of the row and the cofactors are the same as for the matrix $A$ for the same row. Observe that the second row is not used when building the cofactors. Try to do the expanstion for this matrix yourself.
          – A.Γ.
          Jan 4 at 15:00






          @KidCudi Yes, it may take time to understand. The determinant expansion of a matrix is the sum of the element${}_{ij}cdot$the cofactor${}_{ij}$. The expression has the same structure as the determinant expansion along the second row. The red elements must be the elements of the row and the cofactors are the same as for the matrix $A$ for the same row. Observe that the second row is not used when building the cofactors. Try to do the expanstion for this matrix yourself.
          – A.Γ.
          Jan 4 at 15:00






          1




          1




          @KidCudi I have edited the answer, hopefully it is clearer now.
          – A.Γ.
          Jan 4 at 15:13




          @KidCudi I have edited the answer, hopefully it is clearer now.
          – A.Γ.
          Jan 4 at 15:13




          1




          1




          @KidCudi I am saying that the expression in question $a_{11}C_{21}+ldots$ is the determinant expansion of $B$, which is zero. Then the expression is also zero.
          – A.Γ.
          Jan 4 at 15:17




          @KidCudi I am saying that the expression in question $a_{11}C_{21}+ldots$ is the determinant expansion of $B$, which is zero. Then the expression is also zero.
          – A.Γ.
          Jan 4 at 15:17




          1




          1




          I have added a $3$-rd row to your two matrices to make the pattern behind the vdots clearer.
          – darij grinberg
          Jan 4 at 20:26




          I have added a $3$-rd row to your two matrices to make the pattern behind the vdots clearer.
          – darij grinberg
          Jan 4 at 20:26


















          draft saved

          draft discarded




















































          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%2f3061697%2fact-detai%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

          1300-talet

          1300-talet

          Display a custom attribute below product name in the front-end Magento 1.9.3.8