Problem 26 - 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) \(B^{T} A^{T}\) (b) \(C^{T} B^{T}\) (c) \(D^{T} A\)

Step-by-Step Solution

Answer

(a) The expression \(B^T A^T\) is defined, and after computation: \(B^T A^T = \begin{bmatrix} 7 \\ -19 \end{bmatrix}\) (b) The expression \(C^T B^T\) is defined, and after computation: \(C^T B^T = \begin{bmatrix} 36& -64 & 20\\ -4& 1& -3\\ -20& 36 & -16\\ 8& -18 & 4 \end{bmatrix}\) (c) The expression \(D^T A\) is not defined.

Step 1: Transpose of A and B

Matrix A has dimensions 1x3, and its transpose (A^T) will have dimensions 3x1. Matrix B has dimensions 3x2, and its transpose (B^T) will have dimensions 2x3.

Step 2: Transpose of C and D

Matrix C has dimensions 2x4, and its transpose (C^T) will have dimensions 4x2. Matrix D has dimensions 3x3, and its transpose (D^T) will have dimensions 3x3. (a) \(B^TA^T\)

Step 3: Checking if expression (a) is defined

To multiply two matrices, we need the number of columns in the first matrix to be equal to the number of rows in the second matrix. B^T has dimensions 2x3 and A^T has dimensions 3x1. Since the number of columns in B^T (3) is equal to the number of rows in A^T(3), the expression is defined.

Step 4: Computing Expression (a)

Now, let's compute the product of B^T and A^T: \(B^T = \begin{bmatrix} 0 & -7 & -1\\\ -4 & 1 & -3 \end{bmatrix}\) \(A^T = \begin{bmatrix} -3\\\ -1\\\ 6 \end{bmatrix}\) \(B^TA^T = \begin{bmatrix} 0 & -7 & -1\\\ -4 & 1 & -3 \end{bmatrix} \begin{bmatrix} -3\\\ -1\\\ 6 \end{bmatrix} = \begin{bmatrix} 7 \\ -19 \end{bmatrix}\) (b) \(C^TB^T\)

Step 5: Checking if expression (b) is defined

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. C^T has dimensions 4x2, and B^T has dimensions 2x3. Since the number of columns in C^T (2) is equal to the number of rows in B^T (2), the expression is defined.

Step 6: Computing Expression (b)

Now, let's compute the product of C^T and B^T: \(C^T = \begin{bmatrix} -9 & 1\\\ 0 & 1\\\ 3 & 5\\\ -2 & -2 \end{bmatrix}\) \(C^TB^T = \begin{bmatrix} -9 & 1\\\ 0 & 1\\\ 3 & 5\\\ -2 & -2 \end{bmatrix} \begin{bmatrix} 0 & -7 & -1\\\ -4 & 1 & -3 \end{bmatrix} = \begin{bmatrix} 36& -64 & 20\\\ -4& 1& -3\\\ -20& 36 & -16\\\ 8& -18 & 4 \end{bmatrix}\) (c) \(D^TA\)

Step 7: Checking if expression (c) is defined

To multiply two matrices, we need the number of columns in the first matrix equal to the number of rows in the second matrix. D^T has dimensions 3x3, and A has dimensions 1x3. Since the number of columns in D^T (3) is not equal to the number of rows in A (1), the expression is not defined.

Key Concepts

Matrix Transpose

Matrix Transpose is an essential operation in linear algebra that involves flipping a matrix over its diagonal. In simple terms, the rows of the matrix become columns, and the columns become rows. It is denoted by the symbol \( ^T \). For example, if matrix \( A \) is a 1x3 matrix given as \( \begin{bmatrix} -3 & -1 & 6 \end{bmatrix} \), then its transpose, \( A^T \), will be a 3x1 matrix as \( \begin{bmatrix} -3 \ -1 \ 6 \end{bmatrix} \). Knowing how to transpose a matrix is important when performing certain matrix operations, such as matrix multiplication. It can also be a useful property in algorithms that involve symmetric matrices, where the transpose equals the original matrix. Keep in mind:

• The transpose of a transpose matrix returns the original matrix, \( (A^T)^T = A \).
• The transpose of a sum is the sum of the transposes, \( (A+B)^T = A^T + B^T \).
• If you transpose a product of matrices, you reverse the order, \( (AB)^T = B^T A^T \).

Matrix Dimensions

Understanding the dimensions of a matrix is crucial when performing matrix operations such as addition, multiplication, or finding transposes. The dimensions of a matrix are given in terms of rows and columns, denoted as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.In the problem at hand:

• Matrix \( A \) has dimensions 1x3, which means it has 1 row and 3 columns.
• Matrix \( B \) has dimensions 3x2, with 3 rows and 2 columns.
• Matrix \( C \) has dimensions 2x4, implying 2 rows and 4 columns.
• Matrix \( D \) is a 3x3 square matrix consisting of 3 rows and 3 columns.

To perform matrix multiplication, you must ensure the number of columns in the first matrix matches the number of rows in the second matrix. This understanding helps in determining whether a particular matrix operation or multiplication is possible.

Matrix Operations

Matrix operations are the foundations of various linear algebra techniques and are used extensively in different fields such as physics, computer graphics, and statistics. Some common matrix operations include addition, subtraction, multiplication, and finding the transpose.

Matrix Multiplication

Matrix multiplication involves more than simply multiplying corresponding entries. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second. The result of the multiplication will have dimensions that equal the number of rows of the first matrix and columns of the second matrix. For example, multiplying a 2x3 matrix by a 3x2 matrix yields a 2x2 matrix.

Important Tips:

• When you transpose before multiplying, remember to apply the transpose swap property: \( (AB)^T = B^T A^T \).
• Ensure dimension compatibility for multiplication: if you transpose a matrix, its dimensions change and need to be checked again for compatibility.
• Matrix multiplication is associative: \( A(BC) = (AB)C \), but not commutative, which means \( AB eq BA \) generally.

Understanding these core concepts enables you to solve matrix-related problems more effectively and facilitates a deeper comprehension of algorithms and systems dependent on matrix operations.