︠3a42d238-c12f-47aa-9f7c-847cca19df24i︠
%html
<p>A <a href="http://en.wikipedia.org/wiki/Matrix_%28mathematics%29">matrix</a> is a rectangular array of values.</p>
<pre><span>\[<br /> A_<span>{</span><span>m,n</span><span>}</span> = <br /> <span>\begin</span><span>{</span><span><span>bmatrix</span></span><span>}</span><br />  a_<span>{</span><span>1,1</span><span>}</span> <span>&amp;</span> a_<span>{</span><span>1,2</span><span>}</span> <span>&amp;</span> <span>\cdots</span> <span>&amp;</span> a_<span>{</span><span>1,n</span><span>}</span> <span>\\</span><br />  a_<span>{</span><span>2,1</span><span>}</span> <span>&amp;</span> a_<span>{</span><span>2,2</span><span>}</span> <span>&amp;</span> <span>\cdots</span> <span>&amp;</span> a_<span>{</span><span>2,n</span><span>}</span> <span>\\</span><br />  <span>\vdots</span>  <span>&amp;</span> <span>\vdots</span>  <span>&amp;</span> <span>\ddots</span> <span>&amp;</span> <span>\vdots</span>  <span>\\</span><br />  a_<span>{</span><span>m,1</span><span>}</span> <span>&amp;</span> a_<span>{</span><span>m,2</span><span>}</span> <span>&amp;</span> <span>\cdots</span> <span>&amp;</span> a_<span>{</span><span>m,n</span><span>}</span> <br /> <span>\end</span><span>{</span><span><span>bmatrix</span></span><span>}</span><br /><span> </span></span><br /></pre>

<p>Note $\rightarrow$ in the computational world, indexing usually begins at $0$.&nbsp;</p>
<p>However, traditional math texts (such as ours) often begin indexing with $1$.&nbsp; An annoyance, maybe, but a good thing to get used to.</p>
<p>You'll see the use of both both types of indexing often.&nbsp; It's just a matter of learning to pay attention to details.</p>
<pre><span>\[<br /> A_<span>{</span><span>m,n</span><span>}</span> = <br /> <span>\begin</span><span>{</span><span><span>bmatrix</span></span><span>}</span><br />  a_<span>{</span><span>0,0</span><span>}</span> <span>&amp;</span> a_<span>{</span><span>0,1</span><span>}</span> <span>&amp;</span> <span>\cdots</span> <span>&amp;</span> a_<span>{</span><span>0,(n-1)</span><span>}</span> <span>\\</span><br />  a_<span>{</span><span>1,0</span><span>}</span> <span>&amp;</span> a_<span>{</span><span>1,1</span><span>}</span> <span>&amp;</span> <span>\cdots</span> <span>&amp;</span> a_<span>{</span><span>1,(n-1)</span><span>}</span> <span>\\</span><br />  <span>\vdots</span>  <span>&amp;</span> <span>\vdots</span>  <span>&amp;</span> <span>\ddots</span> <span>&amp;</span> <span>\vdots</span>  <span>\\</span><br />  a_<span>{(</span><span>m-1),0</span><span>}</span> <span>&amp;</span> a_<span>{</span><span>(m-1),1</span><span>}</span> <span>&amp;</span> <span>\cdots</span> <span>&amp;</span> a_<span>{(</span><span>m-1),(n-1)</span><span>}</span> <br /> <span>\end</span><span>{</span><span><span>bmatrix</span></span><span>}</span></span></pre>
<p>&nbsp;</p>

<pre>$M = <br />\begin{bmatrix}<br />a &amp; b &amp; c \\<br />d &amp; e &amp; f \\<br />g &amp; h &amp; i<br />\end{bmatrix}<br />$</pre>
<p>Computationally speaking, we can think of a matrix as a list of lists.</p>

︡9645daf5-f5a0-4fa4-b6bf-8d666917d186︡{"html": "<p>A <a href=\"http://en.wikipedia.org/wiki/Matrix_%28mathematics%29\">matrix</a> is a rectangular array of values.</p>\r\n<pre><span>\\[<br /> A_<span>{</span><span>m,n</span><span>}</span> = <br /> <span>\\begin</span><span>{</span><span><span>bmatrix</span></span><span>}</span><br />  a_<span>{</span><span>1,1</span><span>}</span> <span>&amp;</span> a_<span>{</span><span>1,2</span><span>}</span> <span>&amp;</span> <span>\\cdots</span> <span>&amp;</span> a_<span>{</span><span>1,n</span><span>}</span> <span>\\\\</span><br />  a_<span>{</span><span>2,1</span><span>}</span> <span>&amp;</span> a_<span>{</span><span>2,2</span><span>}</span> <span>&amp;</span> <span>\\cdots</span> <span>&amp;</span> a_<span>{</span><span>2,n</span><span>}</span> <span>\\\\</span><br />  <span>\\vdots</span>  <span>&amp;</span> <span>\\vdots</span>  <span>&amp;</span> <span>\\ddots</span> <span>&amp;</span> <span>\\vdots</span>  <span>\\\\</span><br />  a_<span>{</span><span>m,1</span><span>}</span> <span>&amp;</span> a_<span>{</span><span>m,2</span><span>}</span> <span>&amp;</span> <span>\\cdots</span> <span>&amp;</span> a_<span>{</span><span>m,n</span><span>}</span> <br /> <span>\\end</span><span>{</span><span><span>bmatrix</span></span><span>}</span><br /><span> </span></span><br /></pre>\r\n\r\n<p>Note $\\rightarrow$ in the computational world, indexing usually begins at $0$.&nbsp;</p>\r\n<p>However, traditional math texts (such as ours) often begin indexing with $1$.&nbsp; An annoyance, maybe, but a good thing to get used to.</p>\r\n<p>You'll see the use of both both types of indexing often.&nbsp; It's just a matter of learning to pay attention to details.</p>\r\n<pre><span>\\[<br /> A_<span>{</span><span>m,n</span><span>}</span> = <br /> <span>\\begin</span><span>{</span><span><span>bmatrix</span></span><span>}</span><br />  a_<span>{</span><span>0,0</span><span>}</span> <span>&amp;</span> a_<span>{</span><span>0,1</span><span>}</span> <span>&amp;</span> <span>\\cdots</span> <span>&amp;</span> a_<span>{</span><span>0,(n-1)</span><span>}</span> <span>\\\\</span><br />  a_<span>{</span><span>1,0</span><span>}</span> <span>&amp;</span> a_<span>{</span><span>1,1</span><span>}</span> <span>&amp;</span> <span>\\cdots</span> <span>&amp;</span> a_<span>{</span><span>1,(n-1)</span><span>}</span> <span>\\\\</span><br />  <span>\\vdots</span>  <span>&amp;</span> <span>\\vdots</span>  <span>&amp;</span> <span>\\ddots</span> <span>&amp;</span> <span>\\vdots</span>  <span>\\\\</span><br />  a_<span>{(</span><span>m-1),0</span><span>}</span> <span>&amp;</span> a_<span>{</span><span>(m-1),1</span><span>}</span> <span>&amp;</span> <span>\\cdots</span> <span>&amp;</span> a_<span>{(</span><span>m-1),(n-1)</span><span>}</span> <br /> <span>\\end</span><span>{</span><span><span>bmatrix</span></span><span>}</span></span></pre>\r\n<p>&nbsp;</p>\r\n\r\n<pre>$M = <br />\\begin{bmatrix}<br />a &amp; b &amp; c \\\\<br />d &amp; e &amp; f \\\\<br />g &amp; h &amp; i<br />\\end{bmatrix}<br />$</pre>\r\n<p>Computationally speaking, we can think of a matrix as a list of lists.</p>"}︡
︠e7be017c-ffa5-4a4b-af15-d03c098b4f0b︠
var('a b c d e f g h i')

list1 = [a, b, c]
list2 = [d, e, f]
list3 = [g, h, i]

M = [list1, list2, list3]
M
︡3df26d42-29c6-4eb1-9326-bea0e01359c5︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[a, b, c\\right], \\left[d, e, f\\right], \\left[g, h, i\\right]\\right]</span>"}︡
︠08bdf00d-a7a6-4b4f-9fd4-23d14658e9c8︠
for i in [0..2]: M[i]
︡daac8d7e-84b2-4dd7-aba8-e4665d1c72f6︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[a, b, c\\right]</span>"}︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[d, e, f\\right]</span>"}︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[g, h, i\\right]</span>"}︡
︠8c122a03-2666-4f25-b4ab-a1456fbee9cd︠
M[0]
︡e576788c-a4a3-4824-8704-112eb82f1fb7︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[a, b, c\\right]</span>"}︡
︠ed0e4b99-48c4-40c7-b35c-7508fc5b90d0︠
M[0][0]
︡fda5c294-a28a-4e60-8772-af6a19aaec23︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}a</span>"}︡
︠ce420f09-89ee-431a-bc04-8dc575432311︠
M[0][1]
︡1824d30f-230d-46cd-ae36-1701171fc46c︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}b</span>"}︡
︠3cac414e-4358-4b14-9965-c58542ce59a2︠
M[0][2]
︡8141829c-829c-436f-a47a-83eb7c7c369f︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}c</span>"}︡
︠8cd47efb-7e35-40b8-b355-f2944a0b1844︠
M[1]
︡2c45a61b-4374-4277-9999-291d60e920c0︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[d, e, f\\right]</span>"}︡
︠6f3d4066-9811-42e2-be8e-d19d8c01eb3d︠
M[1][0]
︡a4361262-ba87-4f4c-9c78-d6d6e1e3ae65︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}d</span>"}︡
︠e7c76c87-327a-42fb-b1c1-b02fa26fd4de︠
M[1][1]
︡14b00281-1325-4c24-9788-22bf4852a59c︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e</span>"}︡
︠a97d8c86-b00a-4753-b39c-030ef23219d4︠
M[1][2]
︡ec0f29cc-3f16-4a93-a1bf-24c524426302︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}f</span>"}︡
︠28201286-10d7-4ecf-a822-7b6da763292a︠
M[2]
︡3921843d-0431-415a-b5b5-a579962b3a6e︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[g, h, i\\right]</span>"}︡
︠5ffeb733-323e-4066-b1d6-2540261e96d0︠
M[2][0]
︡80d1ca45-c305-487d-82ee-ce235895217e︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}g</span>"}︡
︠63283831-f625-44f1-a3e4-12843f1c12f0︠
M[2][1]
︡39e12877-250c-46b3-be5f-b24fab75c6fa︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}h</span>"}︡
︠add571cf-a78f-486e-b03e-ddf554d3e4dd︠
M[2][2]
︡13f4ed6f-a3dc-4d7c-bae7-3584b9afa976︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}i</span>"}︡
︠ea3193ef-088c-498b-ab73-f70341846c29i︠
%html
<p>Python supports negative indexing:</p>

︡5930d0d4-e34d-4736-9dc8-8e38a912a7aa︡{"html": "<p>Python supports negative indexing:</p>"}︡
︠cad222c8-fbc7-47a6-8913-27f4b909633f︠
M[-1]
︡271ec3b1-2877-4271-aaec-68c3a3a2766e︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[g, h, i\\right]</span>"}︡
︠a37ad88e-f30d-43d2-b6dd-130e663b8832︠
M[-1][-1]
︡f014e975-ee0b-43e8-8cc5-502edef893cc︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}i</span>"}︡
︠84da123a-aa35-4fc1-9be2-8ceca59d7186︠
M[-2]
︡a0127c2a-bde8-49d3-937b-6097c0d51b26︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[d, e, f\\right]</span>"}︡
︠3d91b33f-8c50-4432-9414-38904a6fef9f︠
M[-3]
︡b44a52b3-0614-469e-aacc-61394558d88d︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[a, b, c\\right]</span>"}︡
︠8d6dafad-c74e-4d96-ac79-81b0324454efi︠
%html
<p>Negative indexing can be very useful.</p>
<p>This list of lists we created is structurally a matrix, but it doesn't yet know how to act like a matrix.&nbsp;</p>
<p>It's kind of a dumb matrix.&nbsp;</p>
<p>Sage provides a way to transform a dumb matrix into a smart matrix.<br /> A smart matrix knows how to perform all kinds of useful matrix operations.</p>
<p>To create a smart matrix from a dumb matrix:</p>

︡6bfe39c1-0171-4556-804e-43f548ab09ad︡{"html": "<p>Negative indexing can be very useful.</p>\r\n<p>This list of lists we created is structurally a matrix, but it doesn't yet know how to act like a matrix.&nbsp;</p>\r\n<p>It's kind of a dumb matrix.&nbsp;</p>\r\n<p>Sage provides a way to transform a dumb matrix into a smart matrix.<br /> A smart matrix knows how to perform all kinds of useful matrix operations.</p>\r\n<p>To create a smart matrix from a dumb matrix:</p>"}︡
︠18b4b2ec-63c1-4e24-bafc-c6ffddea80a2︠
M
︡1557b332-3972-4fa8-ae54-c78a99ab447e︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[a, b, c\\right], \\left[d, e, f\\right], \\left[g, h, i\\right]\\right]</span>"}︡
︠b0f81441-4f0a-4f3a-ac5f-04c62b9cd3cd︠
M = matrix(M)
M
︡e7ac010e-0866-4402-b485-6319d9d6a14f︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\na & b & c \\\\\nd & e & f \\\\\ng & h & i\n\\end{array}\\right)</span>"}︡
︠a72aa40a-ff75-489e-a76b-52c93e85fbc4i︠
%html
<p>$M$ is now a smart matrix!&nbsp; A <strong>matrix</strong> object will automatically display itself as a rectangular array rather than as a list of lists.</p>
<p>The indexing we were using works the same as before:</p>

︡34e940ce-8660-4ebd-aeef-4ec02c1129ab︡{"html": "<p>$M$ is now a smart matrix!&nbsp; A <strong>matrix</strong> object will automatically display itself as a rectangular array rather than as a list of lists.</p>\r\n<p>The indexing we were using works the same as before:</p>"}︡
︠b3e76b3a-247c-4657-89a8-e06559e7ede5︠
M[0][0]
︡138368ed-7cc9-459d-b2d6-47fde84afca3︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}a</span>"}︡
︠e299c4a7-d4da-4fa3-ac51-32af172ebd5b︠
M[1][1]
︡388cf09f-9c01-4597-a58e-118f7e7803c4︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e</span>"}︡
︠76ebc21c-013b-466f-ac74-93c5602cd441︠
M[-1][-1]
︡ff2af8a5-ee4c-4998-a4a2-400f915d0c6f︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}i</span>"}︡
︠360a3f93-4766-4e24-aa17-755bffa8836ei︠
%html
<p>However, with a matrix object, we also have access to another kind of indexing:</p>

︡c4595eba-1d88-4565-84a9-e3e4504369fd︡{"html": "<p>However, with a matrix object, we also have access to another kind of indexing:</p>"}︡
︠7654168b-ca93-42e4-8184-5d634b768faa︠
M[1, 2]
︡21d4e4dc-252c-4282-9e66-46ac22414159︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}f</span>"}︡
︠4df384fa-6f1b-4884-a611-225aab0f3228i︠
%html
<p>This kind of indexing allows us to specify submatrices:</p>

︡1c25d5af-0b9a-4380-a7e9-960a6bb67073︡{"html": "<p>This kind of indexing allows us to specify submatrices:</p>"}︡
︠db8d789b-751b-4fb9-b956-00f54297bc63︠
M[[0,1],[1,2]]
︡63523602-0312-4549-899a-db0af7e4dbb1︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\nb & c \\\\\ne & f\n\\end{array}\\right)</span>"}︡
︠6a55f84c-9186-4021-b9b8-bbdd31af6ec8i︠
%html
<p>Now that $M$ is a matrix object, we can do all sorts of useful matrixy kinds of things with it.</p>
<p>To find out what these things are, enter '$M.$' (the letter $M$ followed by a period) in the cell below, if it's not there already, and then press the <strong>TAB</strong> key.&nbsp;</p>
<p>A window will pop up will all kinds of functions.</p>
<p>Wow!&nbsp; (To close the window, press the escape key.)</p>
<p>&nbsp;</p>

︡0fe21fd6-92e4-4562-8adb-b853423ee0b0︡{"html": "<p>Now that $M$ is a matrix object, we can do all sorts of useful matrixy kinds of things with it.</p>\r\n<p>To find out what these things are, enter '$M.$' (the letter $M$ followed by a period) in the cell below, if it's not there already, and then press the <strong>TAB</strong> key.&nbsp;</p>\r\n<p>A window will pop up will all kinds of functions.</p>\r\n<p>Wow!&nbsp; (To close the window, press the escape key.)</p>\r\n<p>&nbsp;</p>"}︡
︠f8738a84-6a95-4ed5-a29f-b48230f18d67︠
M.
︡16abcd4e-53f2-405d-bdc6-d99857ca7d12︡︡
︠8ab2612a-6a9d-4def-8829-a419af0f5a06i︠
%html
<p>We'll be using just a few of those functions.</p>
<p>Here are some examples.</p>
<p>Evaluate the following:</p>

︡4a818f93-c703-40fd-8525-cc1034f513fc︡{"html": "<p>We'll be using just a few of those functions.</p>\r\n<p>Here are some examples.</p>\r\n<p>Evaluate the following:</p>"}︡
︠4a14264c-6aa9-4eb3-be27-ddc38ef0a357︠
transpose(M)
︡aeefce03-a91e-426a-8049-d8d2eab52c48︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\na & d & g \\\\\nb & e & h \\\\\nc & f & i\n\\end{array}\\right)</span>"}︡
︠b8e6d108-d950-4e4e-b0c9-353c1cb876ef︠
det(M)
︡c1ef18eb-e55e-42a9-8e4f-db9155362f81︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(e i - f h\\right)} a - {\\left(b i - c h\\right)} d + {\\left(b f - c e\\right)} g</span>"}︡
︠e5cacc0c-f566-4106-ad1c-9cb3e45ad774︠
M.columns()
︡d2e0dda7-af5d-428e-9344-8594b066b0e1︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(a,d,g\\right), \\left(b,e,h\\right), \\left(c,f,i\\right)\\right]</span>"}︡
︠d935865f-a0f0-4f15-8a5d-71d758366fff︠
M.rows()
︡c944aaa8-187f-4858-b5db-03fb89cd8e23︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(a,b,c\\right), \\left(d,e,f\\right), \\left(g,h,i\\right)\\right]</span>"}︡
︠157b8e14-aad5-42bf-8dea-fea1627b4597︠
var('k')
k*M
︡79611e32-4de1-405d-9834-98857dfa7b05︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\na k & b k & c k \\\\\nd k & e k & f k \\\\\ng k & h k & i k\n\\end{array}\\right)</span>"}︡
︠6af295a8-77aa-4ea9-a6c1-520a5bb23407︠
A = matrix([[a, b], [c, d]])
B = matrix([[e, f], [g, h]])
︡7a057b24-bc77-4e76-a00e-a6178525d47c︡︡
︠7d53a928-2a49-4498-a5f5-d56da8f1d517︠
A
︡8123b70d-1327-47cf-ae90-3975cac6be20︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\na & b \\\\\nc & d\n\\end{array}\\right)</span>"}︡
︠e8ce7383-3d48-4481-b0e1-04c7dade7d94︠
B
︡970e1467-4f3a-4d75-801a-e35b731f1a33︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\ne & f \\\\\ng & h\n\\end{array}\\right)</span>"}︡
︠cb2e9bb4-b947-4311-aa42-d9f6ec8faaa4︠
A*B
︡5566d583-3d3e-4440-adbc-650f56fb33eb︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\na e + b g & a f + b h \\\\\nc e + d g & c f + d h\n\\end{array}\\right)</span>"}︡
︠857feb52-2e11-4e40-8671-0563d999a1f5︠
B*A
︡2c53a293-3bf2-44ea-a863-d6a9363b0880︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\na e + c f & b e + d f \\\\\na g + c h & b g + d h\n\\end{array}\\right)</span>"}︡
︠60ff4993-bcf0-40e1-b25b-c67acf26582a︠
A+B
︡0709833e-e473-4733-b0d6-dd91082648d8︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\na + e & b + f \\\\\nc + g & d + h\n\\end{array}\\right)</span>"}︡
︠bab822ef-2d81-4a30-9444-fd1a57707729︠
M = matrix(10, [1..100])
M
︡53c2edb0-76c1-46cf-a3f1-4aeb1776b6fd︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrrr}\n1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\\\\n31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\\\\n41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50 \\\\\n51 & 52 & 53 & 54 & 55 & 56 & 57 & 58 & 59 & 60 \\\\\n61 & 62 & 63 & 64 & 65 & 66 & 67 & 68 & 69 & 70 \\\\\n71 & 72 & 73 & 74 & 75 & 76 & 77 & 78 & 79 & 80 \\\\\n81 & 82 & 83 & 84 & 85 & 86 & 87 & 88 & 89 & 90 \\\\\n91 & 92 & 93 & 94 & 95 & 96 & 97 & 98 & 99 & 100\n\\end{array}\\right)</span>"}︡
︠089051e7-1cc8-43b7-a5f0-29db727a5322︠
det(M)
︡49907b2a-bd12-4bb4-b4b1-9e1ee604e7f1︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</span>"}︡
︠44c43064-a883-4c8f-a19b-e0b9b85d1243︠
P = matrix(5, [randint(1, 100) for i in range(25)])
P
︡d996c41d-dd5e-4df7-ac42-e4a53571356d︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n55 & 31 & 95 & 60 & 49 \\\\\n50 & 3 & 96 & 33 & 30 \\\\\n25 & 23 & 98 & 4 & 59 \\\\\n65 & 94 & 84 & 63 & 58 \\\\\n97 & 76 & 46 & 92 & 58\n\\end{array}\\right)</span>"}︡
︠c718af24-1ac0-40f7-92d2-d67cc07b5bf3︠
det(P)
︡a2e6eb5d-1057-4be8-b9f7-8077037dfaeb︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}321289209</span>"}︡
︠1646bf14-f451-4363-b7e0-9329751ad326︠
P^-1
︡2029b0ad-e98e-4da2-8c50-a64c9001bd4e︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n-\\frac{1651746}{35698801} & \\frac{1222898}{35698801} & \\frac{159624}{35698801} & -\\frac{98682}{35698801} & \\frac{699213}{35698801} \\\\\n-\\frac{1340426}{107096403} & \\frac{571331}{321289209} & -\\frac{1541492}{321289209} & \\frac{770255}{35698801} & -\\frac{2262455}{321289209} \\\\\n\\frac{113597}{35698801} & \\frac{359866}{35698801} & -\\frac{194211}{35698801} & \\frac{349522}{35698801} & -\\frac{434070}{35698801} \\\\\n\\frac{1694138}{35698801} & -\\frac{838454}{35698801} & -\\frac{536930}{35698801} & -\\frac{186019}{35698801} & -\\frac{265365}{35698801} \\\\\n\\frac{1711586}{107096403} & -\\frac{9754409}{321289209} & \\frac{8668673}{321289209} & -\\frac{826405}{35698801} & \\frac{4866380}{321289209}\n\\end{array}\\right)</span>"}︡
︠cb8a0632-a9ac-4fa8-a3e0-bff6a3565a57︠
P[[1..3], [2..4]]
︡55a36e7e-2893-4625-90b5-e21c1acecd36︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n96 & 33 & 30 \\\\\n98 & 4 & 59 \\\\\n84 & 63 & 58\n\\end{array}\\right)</span>"}︡
︠1eed4cd8-99f9-4d0b-b3b2-e0f21afbd86e︠
def minor(M, row, col):
    rows = range(row)+range(row+1, M.nrows())
    cols = range(col)+range(col+1, M.ncols())
    return M[rows, cols]
︡a4cefb2f-a004-4318-855d-1c944d28c6f2︡︡
︠a82d7a08-9dca-4808-9f06-3b69f1dacae5︠
minor(P, 2, 3)
︡195a4227-2fe7-4138-b990-fdd5d19386ad︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n55 & 31 & 95 & 49 \\\\\n50 & 3 & 96 & 30 \\\\\n65 & 94 & 84 & 58 \\\\\n97 & 76 & 46 & 58\n\\end{array}\\right)</span>"}︡
︠a77cd357-e1a0-446c-ad7d-2da1a60eac23︠
M = matrix(4, [1..16])
M
︡96b2d24f-0ffd-4b6b-9035-542e19e13d1b︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n1 & 2 & 3 & 4 \\\\\n5 & 6 & 7 & 8 \\\\\n9 & 10 & 11 & 12 \\\\\n13 & 14 & 15 & 16\n\\end{array}\\right)</span>"}︡
︠0de59a8a-6361-43a3-947b-8cfc5c9ae366︠
select = [M[1, 2]]
select
︡134ef5b9-3cd5-4334-9a3c-90994a9c6779︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[7\\right]</span>"}︡
︠54a782e3-5040-4769-9455-c744e2935f43︠
M = minor(M, 1, 2)
M
︡213ac60c-91e5-4cdd-91a7-bfd5521c8b0c︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n1 & 2 & 4 \\\\\n9 & 10 & 12 \\\\\n13 & 14 & 16\n\\end{array}\\right)</span>"}︡
︠f785bf5d-cf9d-4807-ad0c-ecc1d71b57c4︠
select += [M[1, 1]]
select
︡00529971-71c1-410e-85fc-a363243f8236︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[7, 10\\right]</span>"}︡
︠08f420eb-1e1c-4fbe-a9a4-b92b9df5a76d︠
M = minor(M, 1, 1)
M
︡e4d243f2-efac-47f2-aafc-859f252ab15e︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n1 & 4 \\\\\n13 & 16\n\\end{array}\\right)</span>"}︡
︠23a9d133-3703-41c7-ac2c-86045ac7abe1︠
select += [M[1, 0]]
select
︡b0f367b2-b6e3-4201-aa73-205b3eeb6111︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[7, 10, 13\\right]</span>"}︡
︠83b6d309-4efd-4359-9687-5f3e474a3bac︠
M = minor(M, 1, 0)
M
︡5a6e1530-4602-4997-817d-c5f46fd27c4b︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r}\n4\n\\end{array}\\right)</span>"}︡
︠e3291c19-281b-400d-9975-651679a0f230︠
select += [M[0, 0]]
select
︡479a1aaa-a6f7-4105-8ca5-2b2e6fe616f4︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[7, 10, 13, 4\\right]</span>"}︡
︠f088c67f-5dea-49f4-ba48-85064fe470c9︠
sum(select)
︡4335f976-91e4-4c05-bb3d-b008e7be0f29︡{"html": "<span class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}34</span>"}︡