Linear Algebra With Applications (classic Version)
5th Edition
ISBN: 9780135162972
Author: BRETSCHER, OTTO
Publisher: Pearson Education, Inc.,
expand_more
expand_more
format_list_bulleted
Videos
Textbook Question
Chapter 2.4, Problem 42E
A square matrix is called a permutation matrix if it contains a I exactly once in each row and in each column,with all other entries being 0. Examples are
Are permutation matrices invertible? If so, is the inversea permutation matrix as well?
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Pls help on both
Calcúlate the cross product of v and w
Consider the following elevation function for a region of irregular terrain:
z(x, y)
=
1
x² + y²
25
Here, z is the elevation of the terrain over a point (x, y) with x and y being the horizontal coordinates. The
region of interest lies between x = 0 and x = 5, and y 0 and y = 5.
Your tasks are the following:
=
1. Analyze how the elevation changes with respect to x and y. To find the elevation changes, calculate the
partial derivatives of the elevation function z with respect to x and
2. Calculate the total volume of soil above the 0-level (z
region of interest.
=
y.
0). To do so, integrate z(x, y) over the whole
Chapter 2 Solutions
Linear Algebra With Applications (classic Version)
Ch. 2.1 - GOAL Use the concept of a linear transformation in...Ch. 2.1 - GOAL Use the concept of a linear transformation in...Ch. 2.1 - GOAL Use the concept of a linear transformation in...Ch. 2.1 - Find the matrix of the linear transformation...Ch. 2.1 - Consider the linear transformation T from 3 to 2...Ch. 2.1 - Consider the transformationT from 2 to 3 given by...Ch. 2.1 - Suppose v1,v2...,vm are arbitrary vectors in n...Ch. 2.1 - Find the inverse of the linear transformation...Ch. 2.1 - In Exercises 9 through 12, decide whether the...Ch. 2.1 - In Exercises 9 through 12, decide whether the...
Ch. 2.1 - In Exercises 9 through 12, decide whether the...Ch. 2.1 - In Exercises 9 through 12, decide whether the...Ch. 2.1 - Prove the following facts: a. The 22 matrix...Ch. 2.1 - a. For which values of the constantk is the matrix...Ch. 2.1 - For which values of the constants a and b is the...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - In Chapter 1, we mentioned that an old German...Ch. 2.1 - Find an nn matrix A such that Ax=3x , for all x in...Ch. 2.1 - Consider the transformation T from 2 to 2...Ch. 2.1 - Consider the transformation T from 2 to 2 that...Ch. 2.1 - In the example about the French coast guard in...Ch. 2.1 - Let T be a linear transformation from 2 to 2 . Let...Ch. 2.1 - Consider a linear transformation T from 2 to 2 ....Ch. 2.1 - The two column vectors v1 and v2 of a 22 matrix A...Ch. 2.1 - Show that if T is a linear transformation from m...Ch. 2.1 - Prob. 40ECh. 2.1 - Prob. 41ECh. 2.1 - When you represent a three-dimensional object...Ch. 2.1 - a. Consider the vector v=[234] . Is the...Ch. 2.1 - The cross product of two vectors in 3 is given by...Ch. 2.1 - Prob. 45ECh. 2.1 - Prob. 46ECh. 2.1 - Prob. 47ECh. 2.1 - Prob. 48ECh. 2.1 - Prove that if A is a transition matrix and x is a...Ch. 2.1 - Prob. 50ECh. 2.1 - Prob. 51ECh. 2.1 - Prob. 52ECh. 2.1 - Prob. 53ECh. 2.1 - Prob. 54ECh. 2.1 - Prob. 55ECh. 2.1 - For each of the, mini-Webs in Exercises 54 through...Ch. 2.1 - Some parking meters in downtown Geneva,...Ch. 2.1 - Prob. 58ECh. 2.1 - Prob. 59ECh. 2.1 - In the financial pages of a newspaper, one can...Ch. 2.1 - Prob. 61ECh. 2.1 - Prob. 62ECh. 2.1 - Prob. 63ECh. 2.1 - Prob. 64ECh. 2.2 - Sketch the image of the standard L under the...Ch. 2.2 - Find the matrix of a rotation through an angle of...Ch. 2.2 - Consider a linear transformation T from 2 to 3 ....Ch. 2.2 - Interpret the following linear transformation...Ch. 2.2 - The matrix [0.80.60.60.8] represents a rotation....Ch. 2.2 - Let L be the line in 3 that consists of all scalar...Ch. 2.2 - Let L be the line in 3 that consists of all scalar...Ch. 2.2 - Interpret the following linear transformation...Ch. 2.2 - Interpret the following linear transformation...Ch. 2.2 - Find the matrix of the orthogonal projection onto...Ch. 2.2 - Refer to Exercise 10. Find the matrix of the...Ch. 2.2 - Consider a reflection matrix A and a vector x in 2...Ch. 2.2 - Suppose a line L in 2 contains the Unit vector...Ch. 2.2 - Suppose a line L in 3 contains the unit vector...Ch. 2.2 - Suppose a line L in 3 contains the unit vector...Ch. 2.2 - Let T(x)=refL(x) be the reflection about the line...Ch. 2.2 - Consider a matrix A of the form A=[abba] , where...Ch. 2.2 - The linear transformation T(x)=[0.60.80.80.6]x is...Ch. 2.2 - Find the matrices of the linear transformations...Ch. 2.2 - Find the matrices of the linear transformations...Ch. 2.2 - Find the matrices of the linear transformations...Ch. 2.2 - Find the matrices of the linear transformations...Ch. 2.2 - Find the matrices of the linear transformations...Ch. 2.2 - Rotations and reflections have two remarkable...Ch. 2.2 - Find the inverse of the matrix [1k01] ,where k is...Ch. 2.2 - a. Find the scaling matrix A that transforms [21]...Ch. 2.2 - Prob. 27ECh. 2.2 - Prob. 28ECh. 2.2 - Prob. 29ECh. 2.2 - Find a nonzero 22 matrix A such that Ax is...Ch. 2.2 - Prob. 31ECh. 2.2 - Consider the rotation matrix D=[cossinsincos] and...Ch. 2.2 - Consider two nonparallel lines L1 and L2 in 2...Ch. 2.2 - One of the five given matrices represents an...Ch. 2.2 - Prob. 35ECh. 2.2 - Prob. 36ECh. 2.2 - Prob. 37ECh. 2.2 - The determinant of a matrix [abcd] is adbc (wehave...Ch. 2.2 - Describe each of the linear transformations...Ch. 2.2 - Prob. 40ECh. 2.2 - Prob. 41ECh. 2.2 - Prob. 42ECh. 2.2 - Prob. 43ECh. 2.2 - A nonzero matrix of the form A=[abba] represents a...Ch. 2.2 - Prob. 45ECh. 2.2 - A nonzero matrix of the form A=[abba] represents a...Ch. 2.2 - Prob. 47ECh. 2.2 - Prob. 48ECh. 2.2 - Prob. 49ECh. 2.2 - Prob. 50ECh. 2.2 - Prob. 51ECh. 2.2 - Prob. 52ECh. 2.2 - Sketch the image of the unit circle under the...Ch. 2.2 - Prob. 54ECh. 2.2 - Prob. 55ECh. 2.2 - Consider an invertible linear transformation T...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - For the matrices A=[ 1 1 1 1],B=[ 1 2 3],C=[ 1 0 1...Ch. 2.3 - Prob. 15ECh. 2.3 - Prob. 16ECh. 2.3 - In the Exercises 17 through 26,find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - Prove the distributive laws for matrices:...Ch. 2.3 - Consider an np matrix A, a pm in matrix B, and...Ch. 2.3 - Consider the matrix D=[cossinsincos] . We know...Ch. 2.3 - Consider the lines P and Q in 2 in the...Ch. 2.3 - Consider two matrices A and B whose product ABis...Ch. 2.3 - Prob. 32ECh. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - In Exercises 43 through 48, find a 22matrix A with...Ch. 2.3 - In Exercises 43 through 48, find a 22matrix A with...Ch. 2.3 - In Exercises 43 through 48, find a 22matrix A with...Ch. 2.3 - In Exercises 43 through 48, find a 22matrix A with...Ch. 2.3 - In Exercises 43 through 48, find a 22matrix A with...Ch. 2.3 - In Exercises 43 through 48, find a 22matrix A with...Ch. 2.3 - In Exercises 49 through 54, consider the matrices...Ch. 2.3 - In Exercises 49 through 54, consider the matrices...Ch. 2.3 - In Exercises 49 through 54, consider the matrices...Ch. 2.3 - Prob. 52ECh. 2.3 - Prob. 53ECh. 2.3 - Prob. 54ECh. 2.3 - In Exercises 55 through 64,find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - Find all upper triangular 22 matrices X such that...Ch. 2.3 - Find all lower triangular 33 matrices X such that...Ch. 2.3 - Prob. 67ECh. 2.3 - Prob. 68ECh. 2.3 - Consider the matrix A2 in Example 4 of Section...Ch. 2.3 - a. Compute A3 for the matrix A in Example 2.3.4....Ch. 2.3 - For the mini-Web in Example 2.3.4, find pages i...Ch. 2.3 - Prob. 72ECh. 2.3 - Prob. 73ECh. 2.3 - Prob. 74ECh. 2.3 - Prob. 75ECh. 2.3 - Prob. 76ECh. 2.3 - Prob. 77ECh. 2.3 - Prob. 78ECh. 2.3 - Prob. 79ECh. 2.3 - Prob. 80ECh. 2.3 - Prob. 81ECh. 2.3 - Prob. 82ECh. 2.3 - Prob. 83ECh. 2.3 - Prob. 84ECh. 2.3 - Prob. 85ECh. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Prob. 16ECh. 2.4 - Prob. 17ECh. 2.4 - Prob. 18ECh. 2.4 - Decide whether the linear transformations in...Ch. 2.4 - Decide whether the linear transformations in...Ch. 2.4 - Which of the functions f from to in Exercises 21...Ch. 2.4 - Which of the functions f from to in Exercises 21...Ch. 2.4 - Which of the functions f from to in Exercises 21...Ch. 2.4 - Which of the functions f from to in Exercises 21...Ch. 2.4 - Which of the (nonlinear) tranformtions from 2to...Ch. 2.4 - Which of the (nonlinear) tranformtions from 2to...Ch. 2.4 - Which of the (nonlinear) tranformtions from 2to...Ch. 2.4 - Find the inverse of the linear transformation...Ch. 2.4 - For which values of the constant k is the...Ch. 2.4 - For which values of the constants h and c is the...Ch. 2.4 - For which values of the constants a, b, and c is...Ch. 2.4 - Find all matrices [abcd] such that adbc=1 and A1=A...Ch. 2.4 - Consider the matrices of the form A=[abba] ,where...Ch. 2.4 - Consider the diagonal matrix A=[a000b000c] . a....Ch. 2.4 - Consider the upper triangular 33 matrix...Ch. 2.4 - To determine whether a square matrix A is...Ch. 2.4 - If A is an invertible matrix and c is a nonzero...Ch. 2.4 - Find A1 for A=[1k01] .Ch. 2.4 - Consider a square matrix that differs from the...Ch. 2.4 - Show that if a square matrix A has two equal...Ch. 2.4 - Which of the following linear transformations T...Ch. 2.4 - A square matrix is called a permutation matrix if...Ch. 2.4 - Consider two invertible nn matrices A and B. Is...Ch. 2.4 - Consider the nn matrix Mn , with n2 , that...Ch. 2.4 - To gauge the complexity of a computational task,...Ch. 2.4 - Consider the linear system Ax=b ,where A is an...Ch. 2.4 - Give an example of a noninvertible function f from...Ch. 2.4 - Consider an invertible linear transformation...Ch. 2.4 - Input-Output Analysis. (This exercise builds on...Ch. 2.4 - This exercise refers to exercise 49a. Consider the...Ch. 2.4 - Prob. 51ECh. 2.4 - Prob. 52ECh. 2.4 - Prob. 53ECh. 2.4 - Prob. 54ECh. 2.4 - Prob. 55ECh. 2.4 - Prob. 56ECh. 2.4 - Prob. 57ECh. 2.4 - Prob. 58ECh. 2.4 - Prob. 59ECh. 2.4 - Prob. 60ECh. 2.4 - Prob. 61ECh. 2.4 - In Exercises 55 through 65, show that the given...Ch. 2.4 - Prob. 63ECh. 2.4 - Prob. 64ECh. 2.4 - Prob. 65ECh. 2.4 - Prob. 66ECh. 2.4 - Prob. 67ECh. 2.4 - For two invertible nnmatrices A and B, determine...Ch. 2.4 - Prob. 69ECh. 2.4 - For two invertible nnmatrices A and B, determine...Ch. 2.4 - Prob. 71ECh. 2.4 - Prob. 72ECh. 2.4 - Prob. 73ECh. 2.4 - Prob. 74ECh. 2.4 - For two invertible nnmatrices A and B, determine...Ch. 2.4 - Find all linear transformations T from 2 to 2...Ch. 2.4 - Prob. 77ECh. 2.4 - Prob. 78ECh. 2.4 - Prob. 79ECh. 2.4 - Consider the regular tetrahedron sketched below,...Ch. 2.4 - Find the matrices of the transformations T and L...Ch. 2.4 - Consider the matrix E=[100310001] and an arbitrary...Ch. 2.4 - Are elementary matrices invertible? If so, is the...Ch. 2.4 - a. Justify the following: If A is an nm in matrix,...Ch. 2.4 - a. Justify the following: If A is an nm...Ch. 2.4 - a. Justify the following: Any invertible matrix is...Ch. 2.4 - Write all possible forms of elementary...Ch. 2.4 - Prob. 88ECh. 2.4 - Prob. 89ECh. 2.4 - Prob. 90ECh. 2.4 - Prob. 91ECh. 2.4 - Show that the matrix A=[0110] cannot be written...Ch. 2.4 - In this exercise we will examine which invertible...Ch. 2.4 - Prob. 94ECh. 2.4 - Prob. 95ECh. 2.4 - Prob. 96ECh. 2.4 - Prob. 97ECh. 2.4 - Prob. 98ECh. 2.4 - Prob. 99ECh. 2.4 - Prob. 100ECh. 2.4 - Prob. 101ECh. 2.4 - Prob. 102ECh. 2.4 - Prob. 103ECh. 2.4 - The color of light can be represented in a vector...Ch. 2.4 - Prob. 105ECh. 2.4 - Prob. 106ECh. 2.4 - Prob. 107ECh. 2.4 - Prob. 108ECh. 2 - The matrix [5665] represents a rotation...Ch. 2 - If A is any invertible nn matrix, then A...Ch. 2 - Prob. 3ECh. 2 - Matrix [1/21/21/21/2] represents a rotation.Ch. 2 - Prob. 5ECh. 2 - Prob. 6ECh. 2 - Prob. 7ECh. 2 - Prob. 8ECh. 2 - Prob. 9ECh. 2 - Prob. 10ECh. 2 - Matrix [k25k6] is invertible for all real numbers...Ch. 2 - There exists a real number k such that the matrix...Ch. 2 - Prob. 13ECh. 2 - Prob. 14ECh. 2 - Prob. 15ECh. 2 - Prob. 16ECh. 2 - Prob. 17ECh. 2 - Prob. 18ECh. 2 - Prob. 19ECh. 2 - Prob. 20ECh. 2 - Prob. 21ECh. 2 - Prob. 22ECh. 2 - Prob. 23ECh. 2 - There exists a matrix A such that [1212]A=[1111] .Ch. 2 - Prob. 25ECh. 2 - Prob. 26ECh. 2 - Prob. 27ECh. 2 - There exists a nonzero upper triangular 22 matrix...Ch. 2 - Prob. 29ECh. 2 - Prob. 30ECh. 2 - Prob. 31ECh. 2 - Prob. 32ECh. 2 - Prob. 33ECh. 2 - If A2 is invertible, then matrix A itself must be...Ch. 2 - Prob. 35ECh. 2 - Prob. 36ECh. 2 - Prob. 37ECh. 2 - Prob. 38ECh. 2 - Prob. 39ECh. 2 - Prob. 40ECh. 2 - Prob. 41ECh. 2 - Prob. 42ECh. 2 - Prob. 43ECh. 2 - Prob. 44ECh. 2 - Prob. 45ECh. 2 - Prob. 46ECh. 2 - Prob. 47ECh. 2 - Prob. 48ECh. 2 - Prob. 49ECh. 2 - Prob. 50ECh. 2 - Prob. 51ECh. 2 - Prob. 52ECh. 2 - Prob. 53ECh. 2 - Prob. 54ECh. 2 - Prob. 55ECh. 2 - Prob. 56ECh. 2 - Prob. 57ECh. 2 - Prob. 58ECh. 2 - Prob. 59ECh. 2 - Prob. 60ECh. 2 - Prob. 61ECh. 2 - For every transition matrix A there exists a...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Similar questions
- math 62arrow_forwardMake up two polynomial functions, f(x) and g(x). • f(x) should be of degree 3 or higher. g(x) should be of degree 4 or higher. • Find f(3) in each of the three ways: substitution, remainder theorem (synthetic division), and long division. You should get the same answer three times for f(3). Find g(-2) once using your choice of the three methods.arrow_forwardere are many real-world situations that exhibit exponential and logarithmic nctions. • Describe two real world scenarios, one exponential and one logarithmic. Do not identify yet whether your scenarios are logarithmic or exponential.arrow_forward
- stacie is a resident at a medical facility you work at. You are asked to chart the amount of solid food that she consumes.For the noon meal today, she ate 1/2 of a 3 ounce serving of meatloaf, 3/4 of her 3 ounce serving of mashed potatoes, and 1/3 of her 2 ounce serving of green beans. Show in decimal form how many ounces of solid food that Stacie consumedarrow_forwardI've been struggling with this because of how close the numbers are together!! I would really appreciate if someone could help me❤️arrow_forwardMatrix MЄ R4×4, as specified below, is an orthogonal matrix - thus, it fulfills MTM = I. M (ELES),- m2,1. We know also that all the six unknowns mr,c are non-negative with the exception of Your first task is to find the values of all the six unknowns. Think first, which of the mr,c you should find first. Next, consider a vector v = (-6, 0, 0, 8) T. What's the length of v, i.e., |v|? Using M as transformation matrix, map v onto w by w = Mv provide w with its numeric values. What's the length of w, especially when comparing it to the length of v? Finally, consider another vector p = ( 0, 0, 8, 6) T. What's the angle between v (from above) and p? Using M as transformation matrix, map p onto q by q = Mp - provide q with its numeric values. What's the angle between w and q, especially when comparing it to the angle between v and p?arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning
Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,
Algebra for College StudentsAlgebraISBN:9781285195780Author:Jerome E. Kaufmann, Karen L. SchwittersPublisher:Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Elements Of Modern Algebra
Algebra
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Cengage Learning,
Algebra for College Students
Algebra
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Cengage Learning
HOW TO FIND DETERMINANT OF 2X2 & 3X3 MATRICES?/MATRICES AND DETERMINANTS CLASS XII 12 CBSE; Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=bnaKGsLYJvQ;License: Standard YouTube License, CC-BY
What are Determinants? Mathematics; Author: Edmerls;https://www.youtube.com/watch?v=v4_dxD4jpgM;License: Standard YouTube License, CC-BY