Problem 27 - Differential Equations and Linear Algebra

Question

Let $$A=\left[\begin{array}{lll} -3 & -1 & 6 \end{array}\right], \quad B=\left[\begin{array}{rr} 0 & -4 \\ -7 & 1 \\ -1 & -3 \end{array}\right]$$ $$C=\left[\begin{array}{rrrr} -9 & 0 & 3 & -2 \\ 1 & 1 & 5 & -2 \end{array}\right], D=\left[\begin{array}{rrr} -2 & 1 & 5 \\ 0 & 0 & 7 \\ 1 & -2 & -1 \end{array}\right]$$ For each item, decide whether or not the given expression is defined. For each item that is defined, compute the result. (a) \(A D^{T}\) (b) \(\left(C^{T} C\right)^{2}\) (c) \(D^{T} B\)

Step-by-Step Solution

Answer

The computed results are as follows: (a) \(A D^T =\begin{bmatrix}-1 & 7 & -9\end{bmatrix}\), (b) \((C^T C)^2 =\begin{bmatrix}6737 & 66 & -1863 & -1332\\ 149 & 29 & 77 & 32\\ 582 & 115 & 753 & 561\\ -506 & -105 & 317 & -188\end{bmatrix}\), (c) \(D^T B =\begin{bmatrix}-2 & 2\\ 9 & 2\\ 26 & 52\end{bmatrix}\).

Step 1: Compute AD^T

Since the number of columns in A matches the number of rows in \(D^T\), the expression is valid. To compute this, we will first find D^T and then multiply A by D^T. \(D^T =\begin{bmatrix}-2 & 0 & 1\\ 1 & 0 & -2\\ 5 & 7 & -1\end{bmatrix}\) Now, we will multiply A by D^T: \(A D^T = \begin{bmatrix}-3 & -1 & 6\end{bmatrix}\times\begin{bmatrix}-2 & 0 & 1\\ 1 & 0 & -2\\ 5 & 7 & -1\end{bmatrix}=\begin{bmatrix}-1 & 7 & -9\end{bmatrix}\) #b) Check dimensions for (C^T C)^2# First, let's check if the expression is valid. Note that the dimensions of C is \(2\times 4\), so the dimensions of C^T is \(4\times 2\).

Step 2: Compute (C^T C)^2

As the number of columns in C^T matches the number of rows in C, the expression is valid. To compute this, we will first find C^T C and square the result. \(C^T =\begin{bmatrix}-9 & 1\\ 0 & 1\\ 3 & 5\\ -2 & -2\end{bmatrix}\) Now, we will multiply C^T by C: \(C^T C = \begin{bmatrix}-9 & 1\\ 0 & 1\\ 3 & 5\\ -2 & -2\end{bmatrix}\times\begin{bmatrix}-9 & 0 & 3 & -2\\ 1 & 1 & 5 & -2\end{bmatrix}=\begin{bmatrix}82 & 1 & -24 & -16\\ 1 & 1 & 5 & -2\\ 10 & 5 & 28 & 2\\ -18 & -2 & -4 & 8\end{bmatrix}\) Then, \((C^T C)^2 = \begin{bmatrix}82 & 1 & -24 & -16\\ 1 & 1 & 5 & -2\\ 10 & 5 & 28 & 2\\ -18 & -2 & -4 & 8\end{bmatrix}^2 = \begin{bmatrix}6737 & 66 & -1863 & -1332\\ 149 & 29 & 77 & 32\\ 582 & 115 & 753 & 561\\ -506 & -105 & 317 & -188\end{bmatrix}\) #c) Check dimensions for D^T B# First, let's check if the expression is valid. We found dimensions of D^T already, which is \(3\times 3\), and dimensions of B is a \(3\times 2\).

Step 3: Compute D^T B

Since the number of columns in D^T matches the number of rows in B, the expression is valid. To compute this, we will multiply D^T by B: \(D^T B = \begin{bmatrix}-2 & 0 & 1\\ 1 & 0 & -2\\ 5 & 7 & -1\end{bmatrix}\times\begin{bmatrix}0 & -4\\ -7 & 1\\ -1 & -3\end{bmatrix}=\begin{bmatrix}-2 & 2\\ 9 & 2\\ 26 & 52\end{bmatrix}\) The results are: \(a)\: A D^T =\begin{bmatrix}-1 & 7 & -9\end{bmatrix}\), \(b)\: (C^T C)^2 =\begin{bmatrix}6737 & 66 & -1863 & -1332\\ 149 & 29 & 77 & 32\\ 582 & 115 & 753 & 561\\ -506 & -105 & 317 & -188\end{bmatrix}\), \(c)\: D^T B =\begin{bmatrix}-2 & 2\\ 9 & 2\\ 26 & 52\end{bmatrix}\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that involves taking the dot product of the rows of the first matrix with the columns of the second matrix. This operation can only be performed if the number of columns in the first matrix is equal to the number of rows in the second matrix. For instance, if matrix A is of dimension \( m \times n \) and matrix B is of dimension \( n \times p \), the resulting matrix, AB, will have a dimension of \( m \times p \).

To perform matrix multiplication:

• First, ensure the matrices are compatible for multiplication (i.e., the number of columns in the first matrix equals the number of rows in the second matrix).
• Next, multiply each element of the rows of the first matrix by the corresponding element of the columns of the second matrix.
• Summing these products gives the elements of the resulting matrix.

Matrix multiplication is not commutative, meaning that the order in which you multiply matrices matters. AB does not necessarily equal BA.

Transpose of Matrix

The transpose of a matrix is a new matrix achieved by swapping its rows and columns. This means that the first row of the original matrix becomes the first column of the new matrix, the second row becomes the second column, and so on. If matrix D has dimensions of \( m \times n \), then the transpose of D, denoted as \( D^T \), will have dimensions \( n \times m \).

Transposing a matrix is useful in various operations and has unique properties:

• When transposing a matrix, the diagonal elements (elements where row equals column) remain unchanged.
• A double transpose of a matrix returns the original matrix \((D^{TT} = D)\).
• Transposition is distributive over addition, meaning \((A + B)^T = A^T + B^T\).
• It reverses the order of multiplication, so \((AB)^T = B^T A^T\).

Understanding transposition is essential in numerous mathematical and engineering applications where data orientation matters.

Matrix Dimensions

Matrix dimensions, or size, refer to a matrix's arrangement defined by the number of rows and columns it contains. This is expressed as \( m \times n \), where \( m \) denotes the number of rows and \( n \) the number of columns.

Knowing a matrix's dimensions is crucial for many matrix operations, such as addition, multiplication, and finding inverses.

• For addition, two matrices must have the same dimensions.
• For multiplication, the number of columns in the first matrix must equal the number of rows in the second.
• A square matrix (where \( m = n \)) has special properties, often being applicable for determinant and inverse calculations.

By examining dimensions, you can immediately verify if certain operations are possible and foresee the dimensions of the resulting matrix. This foundational understanding assists in a more advanced exploration of linear algebra and its applications.