TEST 3
Score: 20/100 1 / 5 answered
(1) 28 mins
Question 2
Given two square matrices size 3 A and B such that |A|=3,|B|=-7. Applying 2 successive row operations on A to obtain A1 as the following: r1↔ r2, (interchanging r1 and r2 ), r2→-8 r2+3 r3 (replac (2024)
`); }); } $('#search-pretype-options').append(prevbooks); }); } function anon_pretype() { let prebooks = null; try { prebooks = JSON.parse(localStorage.getItem('PRETYPE_BOOKS_ANON')); }catch(e) {} if ('previous_books' in prebooks && 'recommended_books' in prebooks) { previous_books = prebooks.previous_books; recommended_books = prebooks.recommended_books; if (typeof PREVBOOKS !== 'undefined' && Array.isArray(PREVBOOKS)) { new_prevbooks = PREVBOOKS; previous_books.forEach(elem => { for (let i = 0; i < new_prevbooks.length; i++) { if (elem.id == new_prevbooks[i].id) { return; } } new_prevbooks.push(elem); }); new_prevbooks = new_prevbooks.slice(0,3); previous_books = new_prevbooks; } if (typeof RECBOOKS !== 'undefined' && Array.isArray(RECBOOKS)) { new_recbooks = RECBOOKS; for (let j = 0; j < new_recbooks.length; j++) { new_recbooks[j].viewed_at = new Date(); } let insert = true; for (let i=0; i < recommended_books.length; i++){ for (let j = 0; j < new_recbooks.length; j++) { if (recommended_books[i].id == new_recbooks[j].id) { insert = false; } } if (insert){ new_recbooks.push(recommended_books[i]); } } new_recbooks.sort((a,b)=>{ adate = new Date(2000, 0, 1); bdate = new Date(2000, 0, 1); if ('viewed_at' in a) {adate = new Date(a.viewed_at);} if ('viewed_at' in b) {bdate = new Date(b.viewed_at);} // 100000000: instead of just erasing the suggestions from previous week, // we just move them to the back of the queue acurweek = ((new Date()).getDate()-adate.getDate()>7)?0:100000000; bcurweek = ((new Date()).getDate()-bdate.getDate()>7)?0:100000000; aviews = 0; bviews = 0; if ('views' in a) {aviews = acurweek+a.views;} if ('views' in b) {bviews = bcurweek+b.views;} return bviews - aviews; }); new_recbooks = new_recbooks.slice(0,3); recommended_books = new_recbooks; } localStorage.setItem('PRETYPE_BOOKS_ANON', JSON.stringify({ previous_books: previous_books, recommended_books: recommended_books })); build_popup(); } } var whiletyping_search_object = null; var whiletyping_search = { books: [], curriculum: [], topics: [] } var single_whiletyping_ajax_promise = null; var whiletyping_database_initial_burst = 0; //number of consecutive calls, after 3 we start the 1 per 5 min calls function get_whiletyping_database() { //gets the database from the server. // 1. by validating against a local database value we confirm that the framework is working and // reduce the ammount of continuous calls produced by errors to 1 per 5 minutes. return localforage.getItem('whiletyping_last_attempt').then(function(value) { if ( value==null || (new Date()) - (new Date(value)) > 1000*60*5 || (whiletyping_database_initial_burst < 3) ) { localforage.setItem('whiletyping_last_attempt', (new Date()).getTime()); // 2. Make an ajax call to the server and get the search database. let databaseUrl = `/search/whiletype_database/`; let resp = single_whiletyping_ajax_promise; if (resp === null) { whiletyping_database_initial_burst = whiletyping_database_initial_burst + 1; single_whiletyping_ajax_promise = resp = new Promise((resolve, reject) => { $.ajax({ url: databaseUrl, type: 'POST', data:{csrfmiddlewaretoken: "MJlqan7YBsfcGGhTk6icwE6YBaaZ32ikZjT45J1nZAoT9wTj7xN4OKnUJNxBTqyT"}, success: function (data) { // 3. verify that the elements of the database exist and are arrays if ( ('books' in data) && ('curriculum' in data) && ('topics' in data) && Array.isArray(data.books) && Array.isArray(data.curriculum) && Array.isArray(data.topics)) { localforage.setItem('whiletyping_last_success', (new Date()).getTime()); localforage.setItem('whiletyping_database', data); resolve(data); } }, error: function (error) { console.log(error); resolve(null); }, complete: function (data) { single_whiletyping_ajax_promise = null; } }) }); } return resp; } return Promise.resolve(null); }).catch(function(err) { console.log(err); return Promise.resolve(null); }); } function get_whiletyping_search_object() { // gets the fuse objects that will be in charge of the search if (whiletyping_search_object){ return Promise.resolve(whiletyping_search_object); } database_promise = localforage.getItem('whiletyping_database').then(function(database) { return localforage.getItem('whiletyping_last_success').then(function(last_success) { if (database==null || (new Date()) - (new Date(last_success)) > 1000*60*60*24*30 || (new Date('2023-04-25T00:00:00')) - (new Date(last_success)) > 0) { // New database update return get_whiletyping_database().then(function(new_database) { if (new_database) { database = new_database; } return database; }); } else { return Promise.resolve(database); } }); }); return database_promise.then(function(database) { if (database) { const options = { isCaseSensitive: false, includeScore: true, shouldSort: true, // includeMatches: false, // findAllMatches: false, // minMatchCharLength: 1, // location: 0, threshold: 0.2, // distance: 100, // useExtendedSearch: false, ignoreLocation: true, // ignoreFieldNorm: false, // fieldNormWeight: 1, keys: [ "title" ] }; let curriculum_index={}; let topics_index={}; database.curriculum.forEach(c => curriculum_index[c.id]=c); database.topics.forEach(t => topics_index[t.id]=t); for (j=0; j
Solutions
Textbooks
`); } function build_solutions() { if (Array.isArray(solution_search_result)) { const viewAllHTML = userSubscribed ? `View All` : ''; var solutions_section = $(`
Solutions ${viewAllHTML}
`); let questionUrl = "/questions/xxx/"; let askUrl = "/ask/question/xxx/"; solution_search_result.forEach((elem) => { let url = ('course' in elem)?askUrl:questionUrl; let solution_type = ('course' in elem)?'ask':'question'; let subtitle = ('course' in elem)?(elem.course??""):(elem.book ?? "")+" "+(elem.chapter?"Chapter "+elem.chapter:""); solutions_section.find('#whiletyping-solutions').append(` ${elem.text} ${subtitle} `); }); $('#search-solution-options').empty(); if (Array.isArray(solution_search_result) && solution_search_result.length>0){ $('#search-solution-options').append(solutions_section); } MathJax.typesetPromise([document.getElementById('search-solution-options')]); } } function build_textbooks() { $('#search-pretype-options').empty(); $('#search-pretype-options').append($('#search-solution-options').html()); if (Array.isArray(textbook_search_result)) { var books_section = $(`
The three elementary row operations are: (Row Swap) Exchange any two rows.(Scalar Multiplication) Multiply any row by a constant.(Row Sum) Add a multiple of one row to another row.
An elementary matrix is always a square matrix. Recall the row operations given in Definition 1.3. 2. Any elementary matrix, which we often denote by E, is obtained from applying one row operation to the identity matrix of the same size.
The three basic elementary operations or transformation of a matrix are: Interchange of any two rows or two columns. Multiplication of row or column by a non-zero number. Multiplication of row or column by a non-zero number and add the result to the other row or column.
Final answer: A square matrix is nonsingular when its determinant is nonzero. If a square matrix can be written as the product of elementary matrices, it means that the matrix can be obtained by performing a sequence of elementary row operations on the identity matrix.
The transpose of a matrix is found by interchanging its rows into columns or columns into rows. The transpose of the matrix is denoted by using the letter “T” in the superscript of the given matrix.
The determinant of a 2x2 matrix A = ⎡⎢⎣abcd⎤⎥⎦ [ a b c d ] is |A| = ad - bc. It is simply obtained by cross multiplying the elements starting from top left and then subtracting the products.
The process is the same for the matrix of any order. We multiply the elements of each row of the first matrix by the elements of each column in the second matrix (element by element) as shown in the image.Finally, we add the products. The result of the product of two 2x2 matrices is again a 2x2 matrix.
Only gradually did the idea of the matrix as an algebraic entity emerge. The term matrix was introduced by the 19th-century English mathematician James Sylvester, but it was his friend the mathematician Arthur Cayley who developed the algebraic aspect of matrices in two papers in the 1850s.
A matrix A of dimension n x n is called invertible if and only if there exists another matrix B of the same dimension, such that AB = BA = I, where I is the identity matrix of the same order. Matrix B is known as the inverse of matrix A. Inverse of matrix A is symbolically represented by A-1.
If the rows of a matrix A have zero elements and columns of matrix B (same order) have zero elements, then the product of matrices A and B will result in a zero matrix.
Thus EA is the result of adding k times row p of A to row q, as required. Every elementary matrix E is invertible, and E−1 is also a elementary matrix (of the same type).
Elementary row operations. Each element in a row can be multiplied by a non-zero constant. It is also known as scaling a row. A row can be replaced by the sum of that row and a multiple of another row.
An elementary row operation is any one of the following moves: Swap: Swap two rows of a matrix.Scale: Multiply a row of a matrix by a nonzero constant.Pivot: Add a multiple of one row of a matrix to another row.
By using only elementary row operations, we do not lose any information contained in the augmented matrix. Our strategy is to progressively alter the augmented matrix using elementary row operations until it is in row echelon form. This process is known as Gaussian elimination.
Introduction: My name is Trent Wehner, I am a talented, brainy, zealous, light, funny, gleaming, attractive person who loves writing and wants to share my knowledge and understanding with you.
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