线性代数英文课件4.3
线性代数 英文讲义
Chapter 1 Matrices and Systems of EquationsLinear systems arise in applications to such areas as engineering, physics, electronics, business, economics, sociology（社会学）, ecology （生态学）, demography(人口统计学), and genetics（遗传学）, etc. §1. Systems of Linear EquationsNew words and phrases in this section:Linear equation 线性方程Linear system，System of linear equations 线性方程组Unknown 未知量Consistent 相容的Consistence 相容性Inconsistent不相容的Inconsistence 不相容性Solution 解Solution set 解集Equivalent 等价的Equivalence 等价性Equivalent system 等价方程组Strict triangular system 严格上三角方程组Strict triangular form 严格上三角形式Back Substitution 回代法Matrix 矩阵Coefficient matrix 系数矩阵Augmented matrix 增广矩阵Pivot element 主元Pivotal row 主行Echelon form 阶梯形1.1 DefinitionsA linear equation (线性方程) in n unknowns（未知量）is1122...n na x a x a x b+++=A linear system of m equations in n unknowns is11112211211222221122...... .........n n n n m m m n n m a x a x a x b a x a x a x b a x a x a x b+++=⎧⎪+++=⎪⎨⎪⎪+++=⎩ This is called a m x n (read as m by n) system.A solution to an m x n system is an ordered n-tuple of numbers （n 元数组）12(,,...,)n x x x that satisfies all the equations.A system is said to be inconsistent （不相容的） if the system has no solutions.A system is said to be consistent （相容的）if the system has at least one solution.The set of all solutions to a linear system is called the solution set（解集）of the linear system.1.2 Geometric Interpretations of 2x2 Systems11112212112222a x a xb a x a x b +=⎧⎨+=⎩ Each equation can be represented graphically as a line in the plane. The ordered pair 12(,)x x will be a solution if and only if it lies on bothlines.In the plane, the possible relative positions are(1) two lines intersect at exactly a point; (The solution set has exactly one element)(2)two lines are parallel; (The solution set is empty)(3)two lines coincide. (The solution set has infinitely manyelements)The situation is the same for mxn systems. An mxn system may not be consistent. If it is consistent, it must either have exactly one solution or infinitely many solutions. These are only possibilities.Of more immediate concerns is the problem of finding all solutions to a given system.1.3 Equivalent systemsTwo systems of equations involving the same variables are said to be equivalent（等价的，同解的）if they have the same solution set.To find the solution set of a system, we usually use operations to reduce the original system to a simpler equivalent system.It is clear that the following three operations do not change the solution set of a system.(1)Interchange the order in which two equations of a system arewritten;(2)Multiply through one equation of a system by a nonzero realnumber;(3)Add a multiple of one equation to another equation. (subtracta multiple of one equation from another one)Remark: The three operations above are very important in dealing with linear systems. They coincide with the three row operations of matrices. Ask a student about the proof.1.4 n x n systemsIf an nxn system has exactly one solution, then operation 1 and 3 can be used to obtain an equivalent “strictly triangular system ”A system is said to be in strict triangular form (严格三角形) if in the k-th equation the coefficients of the first k-1 variables are all zero and the coefficient ofkx is nonzero. (k=1, 2, …,n)An example of a system in strict triangular form:123233331 2 24x x x x x x ++=⎧⎪-=⎨⎪=⎩Any nxn strictly triangular system can be solved by back substitution (回代法).(Note: A phrase: “substitute 3 for x ” == “replace x by 3”)In general, given a system of linear equations in n unknowns, we will use operation I and III to try to obtain an equivalent system that is strictly triangular.We can associate with a linear system an mxn array of numbers whose entries are coefficient of theix ’s. we will refer to this array as thecoefficient matrix (系数矩阵) of the system.111212122212.....................n nm m m n a a a a a a a a a ⎛⎫⎪ ⎪ ⎪ ⎪⎝⎭A matrix （矩阵） is a rectangular array of numbersIf we attach to the coefficient matrix an additional column whose entries are the numbers on the right-hand side of the system, we obtain the new matrix11121121222212n n s m m m na a ab a a a b b a a a ⎛⎫ ⎪ ⎪ ⎪⎝⎭We refer to this new matrix as the augmented matrix （增广矩阵） of a linear system.The system can be solved by performing operations on the augmented matrix. i x ’s are placeholders that can be omitted until the endof computation.Corresponding to the three operations used to obtain equivalent systems, the following row operation may be applied to the augmented matrix.1.5 Elementary row operationsThere are three elementary row operations:(1)Interchange two rows;(2)Multiply a row by a nonzero number;(3)Replace a row by its sum with a multiple of another row.Remark: The importance of these three operations is that they do not change the solution set of a linear system and may reduce a linear system to a simpler form.An example is given here to illustrate how to perform row operations on a matrix.★Example:The procedure for applying the three elementary row operations:Step 1: Choose a pivot element (主元)(nonzero) from among the entries in the first column. The row containing the pivotnumber is called a pivotal row（主行）. We interchange therows (if necessary) so that the pivotal row is the new firstrow.Multiples of the pivotal row are then subtracted form each of the remaining n-1 rows so as to obtain 0’s in the firstentries of rows 2 through n.Step2: Choose a pivot element from the nonzero entries in column 2, rows 2 through n of the matrix. The row containing thepivot element is then interchanged with the second row ( ifnecessary) of the matrix and is used as the new pivotal row.Multiples of the pivotal row are then subtracted form eachof the remaining n-2 rows so as to eliminate all entries belowthe pivot element in the second column.Step 3: The same procedure is repeated for columns 3 through n-1.Note that at the second step, row 1 and column 1 remain unchanged, at the third step, the first two rows and first two columns remain unchanged, and so on.At each step, the overall dimensions of the system are effectively reduced by 1. (The number of equations and the number of unknowns all decrease by 1.)If the elimination process can be carried out as described, we will arrive at an equivalent strictly triangular system after n-1 steps.However, the procedure will break down if all possible choices for a pivot element are all zero. When this happens, the alternative is to reduce the system to certain special echelon form（梯形矩阵）. AssignmentStudents should be able to do all problems.Hand-in problems are: # 7--#11§2. Row Echelon FormNew words and phrases:Row echelon form 行阶梯形Reduced echelon form 简化阶梯形 Lead variable 首变量 Free variable 自由变量Gaussian elimination 高斯消元Gaussian-Jordan reduction. 高斯-若当消元 Overdetermined system 超定方程组 Underdetermined systemHomogeneous system 齐次方程组 Trivial solution 平凡解2.1 Examples and DefinitionIn this section, we discuss how to use elementary row operations to solve mxn systems.Use an example to illustrate the idea.★ Example : Example 1 on page 13. Consider a system represented by the augmented matrix111111110011220031001131112241⎛⎫ ⎪--- ⎪ ⎪-- ⎪- ⎪ ⎪⎝⎭ 111111001120002253001131001130⎛⎫⎪ ⎪ ⎪ ⎪- ⎪ ⎪⎝⎭………..(The details will given in class)We see that at this stage the reduction to strict triangular form breaks down. Since our goal is to simplify the system as much as possible, we move over to the third column. From the example above, we see that the coefficient matrix that we end up with is not in strict triangular form,it is in staircase or echelon form (梯形矩阵).111111001120000013000004003⎛⎫ ⎪ ⎪ ⎪ ⎪- ⎪ ⎪-⎝⎭The equations represented by the last two rows are:12345345512=0 2=3 0=4 03x x x x x x x x x ++++=⎧⎪++⎪⎪⎨⎪-⎪=-⎪⎩Since there are no 5-tuples that could possibly satisfy these equations, the system is inconsistent.Change the system above to a consistent system.111111110011220031001133112244⎛⎫ ⎪--- ⎪ ⎪-- ⎪ ⎪ ⎪⎝⎭ 111111001120000013000000000⎛⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭The last two equations of the reduced system will be satisfied for any 5-tuple. Thus the solution set will be the set of all 5-tuples satisfying the first 3 equations.The variables corresponding to the first nonzero element in each row of the augment matrix will be referred to as lead variable .（首变量） The remaining variables corresponding to the columns skipped in the reduction process will be referred to as free variables （自由变量）.If we transfer the free variables over to the right-hand side in the above system, then we obtain the system:1352435451 2 3x x x x x x x x x ++=--⎧⎪+=-⎨⎪=⎩which is strictly triangular in the unknown 1x 3x 5x . Thus for each pairof values assigned to 2xand4x , there will be a unique solution.★Definition: A matrix is said to be in row echelon form (i) If the first nonzero entry in each nonzero row is 1.(ii)If row k does not consist entirely of zeros, the number of leading zero entries in row k+1 is greater than the number of leading zero entries in row k.(iii) If there are rows whose entries are all zero, they are below therows having nonzero entries.★Definition : The process of using row operations I, II and III to transform a linear system into one whose augmented matrix is in row echelon form is called Gaussian elimination （高斯消元法）.Note that row operation II is necessary in order to scale the rows so that the lead coefficients are all 1.It is clear that if the row echelon form of the augmented matrix contains a row of the form (), the system is inconsistent.000|1Otherwise, the system will be consistent.If the system is consistent and the nonzero rows of the row echelon form of the matrix form a strictly triangular system (the number of nonzero rows<the number of unknowns), the system will have a unique solution. If the number of nonzero rows<the number of unknowns, then the system has infinitely many solutions. (There must be at least one free variable. We can assign the free variables arbitrary values and solve for the lead variables.)2.2 Overdetermined SystemsA linear system is said to be overdetermined if there are more equations than unknowns.2.3 Underdetermined SystemsA system of m linear equations in n unknowns is said to be underdetermined if there are fewer equations than unknowns (m<n). It is impossible for an underdetermined system to have only one solution.In the case where the row echelon form of a consistent system has free variables, it is convenient to continue the elimination process until all the entries above each lead 1 have been eliminated. The resulting reduced matrix is said to be in reduced row echelon form. For instance,111111001120000013000000000⎛⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭ 110004001106000013000000000⎛⎫⎪- ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭Put the free variables on the right-hand side, it follows that12345463x x x x x =-=--=Thus for any real numbersαandβ, the 5-tuple()463ααββ---is a solution.Thus all ordered 5-tuple of the form ()463ααββ--- aresolutions to the system.2.4 Reduced Row Echelon Form★Definition : A matrix is said to be in reduced row echelon form if :(i)the matrix is in row echelon form.(ii) The first nonzero entry in each row is the only nonzero entry in its column.The process of using elementary row operations to transform a matrix into reduced echelon form is called Gaussian-Jordan reduction.The procedure for solving a linear system:(i) Write down the augmented matrix associated to the system; (ii) Perform elementary row operations to reduce the augmented matrix into a row echelon form;(iii) If the system if consistent, reduce the row echelon form into areduced row echelon form. (iv) Write the solution in an n-tuple formRemark: Make sure that the students know the difference between the row echelon form and the reduced echelon form.Example 6 on page 18: Use Gauss-Jordan reduction to solve the system:1234123412343030220x x x x x x x x x x x x -+-+=⎧⎪+--=⎨⎪---=⎩The details of the solution will be given in class.2.5 Homogeneous SystemsA system of linear equations is said to be homogeneous if theconstants on the right-hand side are all zero.Homogeneous systems are always consistent since it has a trivial solution. If a homogeneous system has a unique solution, it must be the trivial solution.In the case that m<n (an underdetermined system), there will always free variables and, consequently, additional nontrivial solution.Theorem 1.2.1 An mxn homogeneous system of linear equations has a nontrivial solution if m<n.Proof A homogeneous system is always consistent. The row echelon form of the augmented matrix can have at most m nonzero rows. Thus there are at most m lead variables. There must be some free variable. The free variables can be assigned arbitrary values. For each assignment of values to the free variables, there is a solution to the system.AssignmentStudents should be able to do all problems except 17, 18, 20.Hand-in problems are 9, 10, 16,Select one problem from 14 and 19.§3. Matrix AlgebraNew words and phrases:Algebra 代数Scalar 数量，标量Scalar multiplication 数乘 Real number 实数 Complex number 复数 V ector 向量Row vector 行向量 Column vector 列向量Euclidean n-space n 维欧氏空间 Linear combination 线性组合 Zero matrix 零矩阵Identity matrix 单位矩阵 Diagonal matrix 对角矩阵 Triangular matrix 三角矩阵Upper triangular matrix 上三角矩阵 Lower triangular matrix 下三角矩阵 Transpose of a matrix 矩阵的转置(Multiplicative ) Inverse of a matrix 矩阵的逆 Singular matrix 奇异矩阵 Singularity 奇异性Nonsingular matrix 非奇异矩阵 Nonsingularity 非奇异性The term scalar (标量，数量) is referred to as a real number (实数) or a complex number （复数）. Matrix notationAn mxn matrix, a rectangular array of mn numbers.111212122212.....................n nm m m n a a a a a a a a a ⎛⎫⎪ ⎪ ⎪ ⎪⎝⎭()ij A a =3.1 VectorsMatrices that have only one row or one column are of special interest since they are used to represent solutions to linear systems.We will refer to an ordered n-tuple of real numbers as a vector （向量）.If an n-tuple is represented in terms of a 1xn matrix, then we will refer to it as a row vector . Alternatively, if the n-tuple is represented by an nx1 matrix, then we will refer to it as a column vector . In this course, we represent a vector as a column vector.The set of all nx1 matrices of real number is called Euclidean n-space (n 维欧氏空间) and is usually denoted by nR.Given a mxn matrix A, it is often necessary to refer to a particular row or column. The matrix A can be represented in terms of either its column vectors or its row vectors.12(a ,a ,,a )n A = ora (1,:)a(2,:)a(,:)A m ⎛⎫ ⎪⎪= ⎪ ⎪⎝⎭3.2 EqualityFor two matrices to be equal, they must have the same dimensions and their corresponding entries must agree★Definition : Two mxn matrices A and B are said to be equal ifij ij a b =for each ordered pair (i, j)3.3 Scalar MultiplicationIf A is a matrix,αis a scalar, thenαA is the mxn matrix formed by multiplying each of the entries of A byα.★Definition : If A is an mxn matrix, αis a scalar, thenαA is themxn matrix whose (i, j) is ij a αfor each ordered pair (i, j) .3.4 Matrix AdditionTwo matrices with the same dimensions can be added by adding their corresponding entries.★Definition : If A and B are both mxn matrices, then the sum A+B is the mxn matrix whose (i,j) entry isij ija b + for each ordered pair (i, j).An mxn zero matrix (零矩阵) is a matrix whose entries are all zero. It acts as an additive identity on the set of all mxn matrices.A+O=O+A=AThe additive of A is (-1)A since A+(-1)A=O=(-1)A+A.A-B=A+(-1)B-A=(-1)A3.5 Matrix Multiplication and Linear Systems3.5.1 MotivationsRepresent a linear system as a matrix equationWe have yet to defined the most important operation, the multiplications of two matrices. A 1x1 system can be writtena xb =A scalar can be treated as a 1x1 matrix. Our goal is to generalize the equation above so that we can represent an mxn system by a single equation.A X B=Case 1: 1xn systems 1122... n n a x a x a x b +++=If we set()12n A a a a =and12n x x X x ⎛⎫ ⎪⎪= ⎪ ⎪⎝⎭, and define1122...n n AX a x a x a x =+++Then the equation can be written as A X b =。
线性代数 英文讲义
Definition
A matrix is said to be in reduced row echelon form if: ⅰ. The matrix is in row echelon form. ⅱ. The first nonzero entry in each row is the only nonzero entry in its column.
n×n Systems Definition
A system is said to be in strict triangular form if in the kth equation the coefficients of the first k-1 variables are all zero and the coefficient of xk is nonzero (k=1, …,n).
1×n matrix
column vector
x1 x2 X x n
n×1 matrix
Definition
Two m×n matrices A and B are said to be equal if aij=bij for each i and j.
1 1
Matrix Multiplication and Linear Systems
Case 1 One equation in Several Unknows
If we let A (a1 a2 an ) and
Example
x1 x2 1 (a ) x1 x2 3 x 2 x 2 2 1 x1 x2 x3 x4 x5 2 (b) x1 x2 x3 2 x4 2 x5 3 x x x 2 x 3x 2 4 5 1 2 3
线性代数 英文讲义
2 Matrix Representations of Linear Transformation
Theorem 4.2.1 If L is a linear transformation mapping Rn
into Rm, there is an m×n matrix A such that L(x)=Ax for each x Rn. In fact, the jth column vector of A is given x∈R by aj=L(ej) j=1, 2, …, n
Theorem 4.1.1 If L: V →W is a linear transformation
and S is a subspace of V, then (1) Ker(L) is a subspace of V. (2) L(S) is a subspace of W.
Example
Let L be the linear operator on R2 defined by
If there is another basis for
R2:
1 1 u1 = ， u 2 = 1 1
2 0 1 2 L(u1)=Au1= 1 1 1 = 2
2 0 1 2 L(u2)=Au2= 1 1 1 = 2
A is the matrix representing L relative to the ordered bases
If A is the matrix representing L with respect to the bases E and F and x=[v]E (the coordinate vector of v with respect to E) y=[w]F (the coordinate vector of w with respect to F) then L maps v into w if and only if A maps x into y.
线性代数(含全部课后题详细答案)4-3PPT课件
目
CONTENCT
录
• 课程介绍与教学目标 • 向量空间与线性变换 • 行列式与矩阵运算 • 特征值与特征向量 • 课后习题详解 • 课程总结与拓展延伸
01
课程介绍与教学目标
线性代数课程简介
线性代数是数学的一个分支， 研究线性方程组、向量空间、 矩阵等概念和性质。
简要介绍数值计算中常用的迭代法、插值 法、逼近法等基本方法，培养学生运用计 算机解决实际问题的能力。
简要介绍数学建模的基本思想和方法，通 过实例展示数学建模在解决实际问题中的 应用和价值。
THANK YOU
感谢聆听
05
课后习题详解
习题类型及解题思路
计算题
主要针对线性代数中的基本运算，如矩阵的加减、数乘和乘法等。解题思路通常是按照运算规则逐步进行，注意保持 矩阵的维度一致。
证明题
主要考察学生对线性代数基本定理和性质的理解和掌握。解题思路一般是从已知条件出发，结合相关定理和性质进行 推导，最终得出结论。
应用题
行列式性质
行列式具有线性性、交换性、倍加性 等基本性质，这些性质在行列式的计 算和证明中起到重要作用。
矩阵运算规则
矩阵加法
两个矩阵相加，要求它们具有相同的行数和列数， 对应元素相加。
矩阵数乘
一个数与矩阵相乘，将该数与矩阵中的每一个元素 相乘。
矩阵乘法
两个矩阵相乘，要求第一个矩阵的列数等于第二个 矩阵的行数，结果矩阵的行数等于第一个矩阵的行 数，列数等于第二个矩阵的列数。
将线性代数的知识应用于实际问题中，如求解线性方程组、矩阵的特征值和特征向量等。解题思路是首 先建立数学模型，将实际问题转化为线性代数问题，然后利用相关知识进行求解。
线性代数英文课件：ch3-1 Elementary Operations
(注：增广矩阵化为最简形时，线性方程组的解亦求出)
下列哪些矩阵是行最简形？
1 0 3 3
1 0
0
1
0
2
?
0 0 1 1
0
1
0 0
0
0
0
0
0
0
0 3
0
2
?
1 1
2
0
1 0
0
1
0 0
0
0
0 3
0
2
?
1 1
0
1Leabharlann 1 0 0 301
0
2
0 0 0 0
√
0
0
0
0
1 0 0 3
I. ri rj : Interchange row i and row j.
II. kri (or k ri ) : Multiply the ith row by a nonzero
.scalar k.
III. ri krj : Add k times the jth row to the ith row (i j)
0
0
Echelon Form Matrix
Definition 4 A matrix is said to be in row echelon
form(行阶梯型矩阵), or simply an echelon matrix, if:
I. The zero rows, if any, are below all nonzero rows and
1. Elementary Operations and Gaussian Elimination Method
The Cramer’s Rule:前提：
线性代数英文ppt5
Ch6_12
Composition of Matrix Transformations
Let T1: Rn → Rm and T2: Rm → Rs be matrix transformations defined by T1(x) = A1x and T2(x) = A2x. We shall now see that T2。 T1 is defined by the product matrix A2A1. T2。T1(x) = T2(T1(x)) = T2(A1x) = A2A1x
- 2 8 and T 4 17 1
e.g.,
1 1 T 2 11 3
Ch6_7
Theorem 6.1
Let A be an mn matrix. Let x be an element of Rn, interpreted as a column matrix. The transformation T: RnRm, defined by T(x)=Ax, is linear. Such a linear transformation is called a matrix transformation.
Example 4
Let D be the operation of taking the derivation. (D is he same as d d . It is a more appropriate notation in this context than .) dx dx D can be interpreted as a mapping of Pn into itself.
线性代数英文ppt3
Examples
(1) V={ …, -3, -1, 1, 3, 5, 7, …} V is not closed under addition because 1+3=4 V.
(2)
Z={ …, -2, -1, 0, 1, 2, 3, 4, …}
Z is closed under addition because for any a, b Z, a + b Z. Z is not closed under scalar multiplication because ½ is a scalar, for any odd a Z, (½)a Z.
Ch04_9
The Complex Vector Space Cn
Let (u1 , ..., un ) be a sequence of n complexnumbers. T heset of all such sequences is denotedC n . Let operat ions of addition and scalar multiplica tion (by a complexscalar c) be defined on C n as follows:
[ f (- f )](x) f ( x) (- f )(x) f ( x) - [ f ( x)] 0 0( x)
Thus [f + (-f )] = 0, -f is the negative of f.
V={ f | f(x)=ax2+bx+c for some a,b,c R}
Ch04_11
Subspaces
In general, a subset of a vector space may or may not satisfy the closure axioms. However, any subset that is closed under both of these operations satisfies all the other vector space properties.
线性代数英文课件：ch4_Review
x1
x3
1 3
x5 ;
x2
x3
2 3
x5 ;
x3
x3 ;
x4
5 3
x5 ;
x5
x5
1 0 1 0 1/3
0
1
1
0
2
/
3
0 0 0 1 5/3
0
0
0
0
0
x1 x2 x3 x4 x5
c1
1
1
1
0
0
c2
0
1 3 2 3
which means
(k k1 knr ) * k11 kn-r n-r 0 (2)
From question(1) we know
*,1, ,n-r are linearly independent,
(k k1 knr ) 0; k1 0, , kn-r 0. k k1 kn-r 0.
1 0 1 0 1/3
0
1
1
0
2
/
3
0 0 0 1 5/3
0 0 0 0 0
x1 x3 1 / 3;
x2
x3
2 / 3;
x3
x3 ;
x4 5 / 3.
x1 1 1 / 3
x2 x3 x4
c
1
1
0
2 0 5
/ /
3
3
,
c
R.
(6)Let matrix A=(1 ,2 ,3 ,4 ,5 )，solve Ax 0.
1, ,n-r are linearly independent.
k1 knr 0,
So *,1, ,n-r are linearly independent.
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x1 x1 x1
+ −
x2 x2
− x3 + x3 − 2x3
= = =
6 2 0
1 1 −1 1 and we found the matrices with coefficient matrix A = 1 −1 1 0 −2 6 1 −1 1 6 −1 1 1 1 A(1) = 2 −1 A(2) = 1 2 A(3) = 1 0 0 −2 1 0 −2 1
Cramer’s Rule Consider a SLE Ax = b in which A is a square matrix of order n with det A = 0. We can form n new n × n matrices by replacing different columns of A by the column vector b. And if we do so, then we can directly find the value of xj in the unique solution to Ax = b using the determinant of one of these new matrices and the determinant of A. A fellow named Cramer developed a rule for doing this. Before we get to the rule, though, we need to define these new matrices and the notation we use to refer to them. Definition: Let Ax = b be any SLE in which A is a square matrix. We define the matrix A(j ) to be the matrix obtained by replacing column j of A with the column vector b.
159
−4 2 −6 −1
−0+0
(−4)(−1) − (−6)(2) = 4 − (−12) = 16
Finally, for A(3) we can expand along row 3 again: 1 1 6 1 6 det A(3) = det 1 −1 2 = 1 det − 0 + 0 = (1)(2) − (−1)(6) = 2 − (−6) = 8 −1 2 1 0 0 So using Cramer’s Rule, we see that the values of x1 , x2 and x3 in the unique solution to the system are: x1 = 16 det A(1) = = 4, det A 4 x2 = det A(2) 16 = = 4, det A 4 x3 = det A(3) 8 = =2 det A 4
1 −1 −1 1 0 −2
6 b= 2 0 instance columns 6 2 0
We form the matrix A(j ) by replacing the j th column of A by the column vector b. So for to form A(1) we write the numbers from b instead of the first column of A, and then write 2 and 3 of A as usual. And so forth. We get: 6 1 −1 1 6 −1 1 1 1 1 A(1) = 2 −1 A(2) = 1 2 A(3) = 1 −1 0 0 −2 1 0 −2 1 0
Unit 12 For A(2), it will be easiest to 1 det A(2) = det 1 1 = zero out column 1 again: 6 −1 1 6 −1 2 1 = det 0 −4 2 = 1 det 0 −2 0 −6 −1
This means that if these determinants are reasonably easy to find, using Cramer’s Rule can be an easier way to find the solution to a SLE than row reducing. (However if the determinants require a lot of work to calculate, then using Cramer’s Rule involves more work than row reducing. So that’s just obnoxious.) For instance determinants of 2 × 2 matrices are always easy to find, so Cramer’s Rule is a reasonably good way to solve a system of 2 equations in 2 unknowns, as long as the coefficient matrix is nonsingular. And if there are 0’s around then sometimes the determinants of larger square matrices are reasonably easy to find. The following examples show how Cramer’s Rule can be used to find the unique solution to a SLE with a nonsingular square coefficient matrix. It’s important to remember, though, that Cramer’s Rule simply doesn’t apply to a system whose coefficient matrix isn’t square, or has determinant 0. Example 12.2. Use Cramer’s Rule to find the unique solution to the SLE in Example 12.1. Solution: We have the SLE:
Theorem 12.1. Cramer’s Rule Let A be any square matrix of order n with det A = 0 and let b be any n × 1 column vector. Then in the unique solution to the system Ax = b, the value of the j th unknown, xj , is given by: xj = det A(j ) det A
Math 1229A/B
Unit 12: Applications of the Determinant
(text reference: Section 4.3)
c V. Olds 2010
Unit 12
157
12
Applications of the Determinant
We shall finish up the course by looking at a couple of other ways that the determinant of a square matrix can be used, that is, a couple of applications of the determinant of a square matrix. The first is a method of finding the solution to the SLE Ax = b when A is a nonsingular (i.e. invertible) square matrix, without row reducing or finding the inverse matrix. Instead, we calculate det A as well as the determinant of certain other matrices obtained from A and b. After that, we will learn how to find the inverse of square matrix A, when it exists, using det A and another matrix which is obtained using the cofactors of A. Again, this gives us another method for doing something which previously we could only do by row reducing.
Now, how do we use these matrices? Recall that as long as det A = 0, the SLE Ax = b has a unique solution. According to Mr. Cramer, the values of the xj ’s in the unique solution to such
1 6 −1 2 0 0
First, we find det A. We can zero out column 1 and then expand along that column: 1 1 −1 1 1 −1 −2 2 1 = det 0 −2 2 = 1 det det A = det 1 −1 −0+0 −1 −1 1 0 −2 0 −1 −1 = (−2)(−1) − (−1)(2) = 2 − (−2) = 4