Systems of Differential Equations
6.1 Review: Matrices
Linear algebra, particularly the study of matrices, is fundamental in understanding and solving systems of differential equations. This section provides a focused overview of the key concepts in matrix theory that are essential for this purpose.
A. Matrix Definition and Notation
A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries. A matrix is typically denoted by a capital letter (e.g., A, B, C). The element in the [asciimath]i[/asciimath]-th row and [asciimath]j[/asciimath]-th column of a matrix A is denoted as [asciimath]a_(ij)[/asciimath]. The dimensions of a matrix are given as [asciimath]"rows" xx "columns"[/asciimath]. For example, a matrix A with [asciimath]m[/asciimath] rows and [asciimath]n[/asciimath] columns is an [asciimath]mxxn[/asciimath] matrix.
[asciimath]A = [[a_11, a_12 ,..., a_(1n)],[a_21,a_22, ..., a_(2n)], [vdots,vdots ,ddots, vdots] ,[a_(m1), a_(m2) , ..., a_(mn)] ]_(mxxn)[/asciimath]
B. Special Matrices
A row matrix has only one row and multiple columns, while a column matrix has one column and multiple rows. These are also known as row vectors and column vectors, respectively.
[asciimath]x = [[a_11, a_12,...,a_(1n)]]_(1xxn)[/asciimath] [asciimath]y = [[a_11],[a_(21)], [vdots] ,[a_(m1)] ]_(mxx1)[/asciimath]
A matrix with the same number of rows and columns is called a square matrix. For example, matrix B is an [asciimath]nxxn[/asciimath] square matrix.
[asciimath]B = [[a_11, a_12 ,..., a_(1n)],[a_21,a_22, ..., a_(2n)], [vdots,vdots ,ddots, vdots] ,[a_(n1), a_(n2) , ..., a_(n\n)] ]_(nxxn)[/asciimath]
In a diagonal matrix, the elements outside the main diagonal are all zero. The main diagonal is the set of elements [asciimath]a_(ij)[/asciimath] where [asciimath]i=j[/asciimath]. For example, matrix C is an [asciimath]nxxn[/asciimath] diagonal matrix.
[asciimath]C = [[a_11, 0 ,..., 0],[0,a_22, ..., 0], [vdots, vdots,ddots, vdots] ,[0, 0 , ..., a_(n\n)] ]_(nxxn)[/asciimath]
The identity matrix is a special type of diagonal matrix where all the elements on the main diagonal are 1. It is denoted as [asciimath]I[/asciimath] or [asciimath]I_n[/asciimath] to indicate its size ([asciimath]nxxn[/asciimath]).
[asciimath]I_n = [[1, 0 ,..., 0],[0,1, ..., 0], [vdots, vdots,ddots, vdots] ,[0, 0 , ..., 1] ]_(nxxn)[/asciimath]
The zero matrix is a matrix in which all elements are zero. It is denoted by [asciimath]0_(mxxn)[/asciimath]to indicate its dimensions.
[asciimath]0_(mxxn) = [[0, 0 ,..., 0],[0,0, ..., 0], [vdots, vdots,ddots, vdots] ,[0, 0 , ..., 0] ]_(mxxn)[/asciimath]
C. Matrix Operation
Matrix Addition and Subtraction
Matrix addition and subtraction are elementary operations where matrices of the same dimension are added or subtracted element by element. If [asciimath]A=[a_(ij)][/asciimath] and [asciimath]B=[b_(ij)][/asciimath] are matrices of the same size, their sum [asciimath]C=A+B[/asciimath] is a matrix where each element [asciimath]c_(ij)=a_(ij)+b_(ij)[/asciimath].
These operations are commutative (i.e., [asciimath]A+B=B+A[/asciimath]) and associative (i.e., [asciimath](A+B)+C=A+(B+C)[/asciimath]).
Scalar Multiplication
Scalar multiplication involves multiplying every element of a matrix by a scalar (a constant number). If [asciimath]k[/asciimath] is a scalar and [asciimath]A=[a_(ij)][/asciimath], then [asciimath]kA[/asciimath] is a matrix where each element is [asciimath]ka_(ij)[/asciimath].
Scalar multiplication is distributive over matrix addition or subtraction (i.e., [asciimath]k(A+B)=kA+kB[/asciimath] ) and associative with respect to the multiplication of scalars (i.e., [asciimath]k(lA)=(kl)A[/asciimath]).
Find matric [asciimath]C[/asciimath] where [asciimath]C=3A-B[/asciimath] given matrices [asciimath]A[/asciimath] and [asciimath]B[/asciimath] .
[asciimath]A=[[-1,3],[0,7]][/asciimath] and [asciimath]B=[[2,-4],[-2,5]][/asciimath]
Show/Hide Solution
Matrices [asciimath]A[/asciimath] and [asciimath]B[/asciimath] are the same size and thus can be subtracted.
[asciimath]C=3A-B[/asciimath]
[asciimath]C=3[[-1,3],[0,7]] -[[2,-4],[-2,5]][/asciimath]
We first multiply all entries of matrix A by 3.
[asciimath]=[[-3,9],[0,21]] -[[2,-4],[-2,5]][/asciimath]
We then subtract the corresponding entries.
[asciimath]=[[-3-2,9-(-4)],[0-(-2),21-5]][/asciimath]
[asciimath]C=[[-5,13],[2,16]][/asciimath]
Try an Example
Matrix Multiplication
Matrix multiplication is only possible when the number of columns in the first matrix matches the number of rows in the second matrix. Consider two matrices [asciimath]A_(mxxn)[/asciimath] and [asciimath]B_(nxxp)[/asciimath]. The product of these matrices is a new matrix [asciimath]C_(mxxp)[/asciimath], where the dimension of C is [asciimath]mxxp[/asciimath]. Each element of C is computed by taking the dot product of a corresponding row from A and a column from B. This computation for each element in the [asciimath]i[/asciimath]-th row and [asciimath]j[/asciimath]-th column of C is given by the formula
[asciimath]c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{i\n}b_{nj}[/asciimath] (6.1.1)
Matrix multiplication is associative, meaning [asciimath](AB)C = A(BC)[/asciimath]. It is also distributive over addition, which implies [asciimath]A(B + C) = AB + AC[/asciimath]. However, it is not commutative, meaning [asciimath]AB[/asciimath] may not equal [asciimath]BA[/asciimath]).
Special cases in matrix multiplication include interactions with identity and zero matrices. Multiplying any matrix by an identity matrix of appropriate size leaves the matrix unchanged (i.e., [asciimath]AI = IA = A[/asciimath]). Any matrix multiplied by a zero matrix results in a zero matrix of appropriate dimensions.
Compute matrix [asciimath]C=AB[/asciimath] given
[asciimath]A=[[1,4,-1],[2,0,-5]][/asciimath] and [asciimath]B=[[-7,3,-1,0],[-5,1,4,3],[0,-2,1,2]].[/asciimath]
Show/Hide Solution
To compute the product of matrices A and B, [asciimath]AB[/asciimath], we first verify that multiplication is possible. Matrix A has dimensions [asciimath]2xx3[/asciimath], and matrix B has dimensions [asciimath]3xx4[/asciimath]. Since the number of columns in A (3) matches the number of rows in B (3), multiplication can be performed. The resulting matrix C will have dimensions [asciimath]2xx4[/asciimath].
We compute each entry of matrix C using Equation 6.1.1:
[asciimath]c_{11} = (1)(-7)+(4)(-5)+(-1)(0)=-27[/asciimath]
[asciimath]c_{12} = (1)(3)+(4)(1)+(-1)(-2)=9[/asciimath]
[asciimath]c_{13} = (1)(-1)+(4)(4)+(-1)(1)=14[/asciimath]
[asciimath]c_{14} = (1)(0)+(4)(3)+(-1)(2)=10[/asciimath]
[asciimath]c_{21} = (2)(-7)+(0)(-5)+(-5)(0)=-14[/asciimath]
[asciimath]c_{22} = (2)(3)+(0)(1)+(-5)(-2)=16[/asciimath]
[asciimath]c_{23} = (2)(-1)+(0)(4)+(-5)(1)=-7[/asciimath]
[asciimath]c_{24} = (2)(0)+(0)(3)+(-5)(2)=-10[/asciimath]
Therefore, the resulting matrix C is
[asciimath]C=[[-27,9,14,10],[-14,16,-7,-10]][/asciimath]
Try an Example
D. Matrix Determinant
The determinant is a scalar value that is associated with every square matrix. It provides critical information about the matrix, such as its invertibility. The determinant of a matrix A is denoted as
[asciimath]det(A)=|A|[/asciimath]
For a [asciimath]2xx2[/asciimath] matrix, the determinant is calculated as
[asciimath]|[a_11,a_12],[a_21,a_22]|=a_11a_22-a_21a_12[/asciimath](6.1.2)
For larger square matrices, the determinant is typically calculated using the method of cofactor expansion. For instance, the determinant of a [asciimath]3xx3[/asciimath] matrix can be computed by expanding along any row or column. Expanding along the first row, the formula is
[asciimath]|[a_11,a_12,a_13],[a_21,a_22,a_23] ,[a_31,a_32,a_33] |=[/asciimath] [asciimath]a_11|[a_22,a_23],[a_32,a_33]|-[/asciimath] [asciimath]a_12|[a_21,a_23],[a_31,a_33]|+[/asciimath] [asciimath]a_13|[a_21,a_22],[a_31,a_32]|[/asciimath](6.1.3)
Another approach to compute determinants, especially for large matrices, is to use row reduction to transform the matrix into an upper triangular form. The determinant is then the product of the diagonal elements.
Find the determinant of the given matrices.
[asciimath]A=[[-5,-1],[3,2]][/asciimath] and [asciimath]B= [[2,4,7],[5,-3,8],[0,-1,3] ][/asciimath]
Show/Hide Solution
To find the determinant of matrix A, we use Formula 6.1.2.
[asciimath]|A|=[/asciimath] [asciimath]|[-5,-1],[3,2]|=(-5)(2)-(3)(-1)=-7[/asciimath]
To find the determinant of matrix B, we use Formula 6.1.3.
[asciimath]|B|=|[2,4,7],[5,-3,8] ,[0,-1,3] |=[/asciimath] [asciimath]2|[-3,8],[-1,3]|-[/asciimath] [asciimath]4|[5,8],[0,3]|+[/asciimath] [asciimath]7|[5,-3],[0,-1]|[/asciimath]
[asciimath]=2((-3)(3)-(-1)(8))-4((5)(3)-(0)(8))+[/asciimath] [asciimath]7((5)(-1)-(0)(-3))[/asciimath]
[asciimath]=2(-1)-4(15)+7(-5)[/asciimath]
[asciimath]=-97[/asciimath]
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E. Matrix Inverse
The inverse of a square matrix [asciimath]A[/asciimath], denoted as [asciimath]A^(-1)[/asciimath], is a matrix that, when multiplied with [asciimath]A[/asciimath], yields the identity matrix.
[asciimath]A A^-1=A^-1 A=I_n[/asciimath]
One common method to find a matrix inverse is to use the adjugate and determinant. The formula is
[asciimath]A^{-1} = \frac{1}{det(A)} \text{adj}(A)[/asciimath]
where [asciimath]"adj"(A)[/asciimath] is the adjugate of [asciimath]A[/asciimath], calculated from the cofactors of [asciimath]A[/asciimath]. This method involves computing the determinant and then the cofactor matrix, which is then transposed to get the adjugate matrix. For a [asciimath]2xx2[/asciimath] matrix [asciimath]A=[[a_11,a_12],[a_21,a_22]][/asciimath], the inverse is given by
[asciimath]A^-1=1/(det(A))[[a_22,-a_12],[-a_21,a_11]][/asciimath](6.1.4)
Another method for finding the inverse is the row reduction method, which involves augmenting the matrix [asciimath]A[/asciimath] with the identity matrix
[asciimath][A|I][/asciimath]
and then performing row operations to transform [asciimath]A[/asciimath] into the identity matrix. The operations that transform A into [asciimath]I[/asciimath] will transform the augmented identity matrix into [asciimath]A^{-1}[/asciimath].
[asciimath][I|A^-1][/asciimath]
This method is particularly useful for numerical calculations and for larger matrices.
If a matrix is invertible, its inverse is unique. A square matrix is invertible if and only if it is nonsingular, meaning its determinant is not zero. If the determinant of a matrix is zero, the matrix does not have an inverse, and it is referred to as a singular matrix.
Find the inverse of matrix A, provided it exists.
[asciimath]A=[[4,6],[2,6]][/asciimath]
Show/Hide Solution
We first find the determinant of A to determine if it has an inverse.
[asciimath]|A|=(4)(6)-(2)(6)=12[/asciimath]
The determinant is nonzero, so the inverse exists. For a [asciimath]2xx2[/asciimath] matrix, the cofactor approach, Formula 6.1.4, is fairly simple.
[asciimath]A^-1=1/12[[6,-6],[-2,4]][/asciimath]
[asciimath]=[[1/2,-1/2],[-1/6,1/3]][/asciimath]
Try an Example
Find the inverse of matrix A, provided it exists.
[asciimath]A=[[1,0,3],[-3,1,-9],[0,2,1]][/asciimath]
Show/Hide Solution
To find the inverse of a [asciimath]3xx3[/asciimath] matrix, the row reduction method is more straightforward. To find the inverse of matrix [asciimath]A[/asciimath] using the row reduction method, we start by forming an augmented matrix with matrix [asciimath]A[/asciimath] and the [asciimath]3xx3[/asciimath] identity matrix [asciimath]I_3[/asciimath]. The goal is to use row operations to transform the left side of the augmented matrix (the first three columns) into the identity matrix.
[asciimath][A|I]=[/asciimath] [asciimath][[1,0,3,|,1,0,0],[-3,1,-9,|,0,1,0],[0,2,1,|,0,0,1]][/asciimath]
Apply the row operation:
[asciimath]R2=R2+3R1[/asciimath] (Add 3 times the first row to the second row):
[asciimath][[1,0,3,|,1,0,0],[0,1,0,|,3,1,0],[0,2,1,|,0,0,1]][/asciimath]
[asciimath]R3=R3-2R2[/asciimath] (Subtract 2 times the second row from the third row):
[asciimath][[1,0,3,|,1,0,0],[0,1,0,|,3,1,0],[0,0,1,|,-6,-2,1]][/asciimath]
[asciimath]R1=R1-3R3[/asciimath] (Subtract 3 times the third row from the first row):
[asciimath][[1,0,0,|,19,6,-3],[0,1,0,|,3,1,0],[0,0,1,|,-6,-2,1]][/asciimath]
Since we have successfully transformed the left side of the augmented matrix into the identity matrix, the inverse of matrix A exists and is given by the right side of the augmented matrix:
[asciimath]A^-1=[[19,6,-3],[3,1,0],[-6,-2,1]][/asciimath]
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F. Matrix Calculus
Differentiation and integration of matrices are important in the context of systems of linear differential equations, particularly in finding the solution to nonhomogeneous systems.
Matrix Differentiation
Differentiating a matrix with function entries involves taking the derivative of each element of the matrix individually. Consider matrix [asciimath]A(t)[/asciimath] whose entries are a function of [asciimath]t[/asciimath].
[asciimath]A(t) = [[a_11(t), a_12(t) ,..., a_(1n)(t)],[a_21(t),a_22(t), ..., a_(2n)(t)], [vdots,vdots ,ddots, vdots] ,[a_(m1)(t), a_(m2)(t) , ..., a_(mn)(t) ] ][/asciimath]
The derivative of [asciimath]A(t)[/asciimath] with respect to [asciimath]t[/asciimath], denoted as [asciimath]A'(t)[/asciimath] or [asciimath](dA)/(dt)[/asciimath], is a matrix of the same size where each entry is the derivative of the corresponding entry of [asciimath]A(t)[/asciimath].
[asciimath]A'(t) = [[a'_11(t), a'_12(t) ,..., a'_(1n)(t)],[a'_21(t),a'_22(t), ..., a'_(2n)(t)], [vdots,vdots ,ddots, vdots] ,[a'_(m1)(t), a'_(m2)(t) , ..., a'_(mn)(t)] ][/asciimath]
The standard rules of differentiation, including the product rule, quotient rule, and chain rule, apply to each element of the matrix.
Matrix Integration
Integrating a matrix with function entries is similar to differentiation and is done element-wise. The integral of a matrix [asciimath]A(t)[/asciimath] over a variable [asciimath]t[/asciimath] is a matrix of the same size where each element is the integral of the corresponding element of [asciimath]A(t)[/asciimath].
[asciimath]intA(t)dt = [[inta_11(t)dt, inta_12(t)dt ,..., inta_(1n)(t)dt],[inta_21(t)dt,inta_22(t)dt, ..., inta_(2n)(t)dt], [vdots,vdots ,ddots, vdots] ,[inta_(m1)(t)dt, inta_(m2)(t)dt , ..., inta_(mn)(t)dt ] ][/asciimath]
Evaluate the integral of matrix [asciimath]A(t)[/asciimath] with respect to [asciimath]t[/asciimath].
[asciimath]A(t) = [(4 e^(4t) , 3te^(-t^2)) , (t^2cos(-3t^3) , -5t^7)][/asciimath]
Show/Hide Solution
The integral of matrices is an element-wise operation.
[asciimath]intA(t)dt=[intA_(ij)(t)dt][/asciimath]
[asciimath]=[(int4 e^(4t) dt, int3te^(-t^2)dt) , (intt^2cos(-3t^3)dt , int-5t^7dt)][/asciimath]
[asciimath]=[(e^(4t) , -3/2e^(-t^2)) , (-1/9sin(-3t^3) , -5/8t^8)][/asciimath]
Try an Example
Section 6.1 Exercises
- Given [asciimath]A = [(0,-3,3),(3,4,-3)][/asciimath] and [asciimath]B = [(-4,0,-4),(-3,0,0)][/asciimath], find matrix [asciimath]C=2 A -4 B[/asciimath].
Show/Hide Answer
[asciimath]C=[(16,-6,22),(18,8,-6)][/asciimath]
- Find the inverse of [asciimath]A = [(-8,3),(21,-8)][/asciimath].
Show/Hide Answer
[asciimath]A^-1=[(-8,-3),(-21,-8)][/asciimath]
- Find the inverse of [asciimath]A = [(3,2,0),(1,1,0),(-2,-4,1)].[/asciimath]
Show/Hide Answer
[asciimath]A^-1=[(1,-2,0),(-1,3,0),(-2,8,1)][/asciimath]
- Given the matrices [asciimath]A=[(5,-5),(-2,-4)][/asciimath] and [asciimath]B=[(-2,1),(4,4)][/asciimath] , find their multiplication [asciimath]AB[/asciimath].
Show/Hide Answer
[asciimath]AB=[(-30,-15),(-12,-18)][/asciimath]
- Given the matrix
[asciimath]A(t) = [(-2 e^(4t) , 3te^(-3t^2)) , (-5tsin(4t^2) , -7t^-5)][/asciimath]
Evaluate the integral of [asciimath]A[/asciimath] with respect to [asciimath]t[/asciimath].
Show/Hide Answer
[asciimath]intAdt=[/asciimath] [asciimath][(-1/2 e^(4t) , -1/2e^(-3t^2)) , (5/8cos(4t^2) , 7/4t^-4)][/asciimath]
