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Chapter 9: Problem 20

Suppose that \(A\) is an \(m \times k\) matrix and \(B\) is a \(k \times n\) matrix. Show that $$ (A B)^{\prime}=B^{\prime} A^{\prime} $$

Short Answer

Expert verified
The transpose of a product of matrices follows the rule \((A B)' = B' A'\), and both \((AB)'\) and \(B'A'\) have dimensions \(n \times m\).

Step by step solution

01

Understand Matrix Transposition

The transpose of a matrix, denoted as 7"), is obtained by swapping its rows with columns. For example, if matrix \(A\) is \(3 \times 2\), then \(A'\) would be a \(2 \times 3\) matrix.
02

Transpose of a Product

When dealing with the transpose of the product of two matrices, the rule is that \((A B)' = B' A'\). The order of multiplication reverses as well as each individual matrix being transposed.
03

Dimensional Consistency

Ensure dimensions are consistent. Matrix \(A\) is \(m \times k\) and \(B\) is \(k \times n\), so their product \(AB\) is \(m \times n\). The transpose \((AB)'\) thus is \(n \times m\).
04

Transpose Each Matrix Individually

\(B'\), being the transpose of \(B\), will be \(n \times k\), and \(A'\), the transpose of \(A\), will be \(k \times m\). Their product \(B'A'\) is \(n \times m\), matching the dimensions of \((AB)'\).
05

Establish the Relationship

Understand that matrix transposition respects the reversed order of multiplication and corresponding dimension changes, confirming that \((AB)' = B'A'\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Multiplication
Matrix multiplication involves calculating the dot product of rows and columns between two matrices. Ensure you multiply matrix
  • "A", which has dimensions of \(m \times k\), by matrix "B" with dimensions \(k \times n\).
  • The result, known as matrix "C" or "AB", will have dimensions of \(m \times n\).
The key rules for matrix multiplication include:
1. Matching of inner dimensions: the number of columns in the first matrix should equal the number of rows in the second.
2. Each element in the resulting matrix is obtained by taking a dot product of the corresponding row from the first matrix and column from the second matrix.To calculate: - For an element \(c_{ij}\) in matrix C, sum up the products of the elements of the \(i^{th}\) row of A and the \(j^{th}\) column of B. Good attention to detail ensures consistent and error-free calculations.
Matrix Dimensions
Understanding matrix dimensions is crucial for matrix operations like addition, multiplication, and transposition.When we refer to matrix dimensions, we're talking about its rows and columns, structured as \(m \times n\), where "m" is the number of rows and "n" is the number of columns. It鈥檚 akin to describing a rectangle by its height and width, only now it's all about aligning mathematical operations appropriately.Key points about dimensions are:
  • Matrix operation eligibility depends on proper dimension alignment. For example, you can't add or subtract matrices unless they share identical dimensions.
  • In multiplication, the internal dimensions (thus the order of the first matrix鈥檚 columns and the second matrix's rows) must match for the operation to be valid.
  • Dimensions also determine the size of any resulting matrix and influence other matrix operations like the ability to find a determinant or inverse (only square matrices have these).
Careful attention to dimensions helps avoid missteps in computational problems.
Transpose of a Matrix
The transpose of a matrix involves flipping it along its diagonal. This means turning its rows into columns and vice versa.If you have a matrix \(A\) with dimensions \(m \times n\), then the transpose of \(A\), denoted \(A'\), will be \(n \times m\).Important aspects to remember about matrix transposition:
  • Transposing twice will revert the matrix to its original form. Thus, \((A')' = A\).
  • The operation is commutative within multiplication, as seen in product transposition: \((AB)' = B'A'\). This means first transpose each matrix and then reverse the multiplication order compared to the original product.
  • Matrix transposition is also utilized in various mathematical strategies to simplify solving linear equations or determining orthogonal potential in matrices.
Understanding and using the transpose effectively can streamline matrix-related computations, reinforcing its significance in both theoretical and applied linear algebra contexts.

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Most popular questions from this chapter

Solve each system of linear equations. $$ \begin{array}{rr} x+4 y-3 z= & -13 \\ 2 x-3 y+5 z= & 18 \\ 3 x+y-2 z= & 1 \end{array} $$

Zach wants to buy fish and plants for his aquarium. Each fish costs \(\$ 2.30 ;\) each plant costs \(\$ 1.70 .\) He buys a total of 11 items and spends a total of \(\$ 21.70 .\) Set up a system of linear equations that will allow you to determine how many fish and how many plants Zach bought, and solve the system.

$$ A=\left[\begin{array}{rr} -1 & 1 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -2 \\ 3 & 2 \end{array}\right] $$ Find the inverse (if it exists) of \(B\).

Three different species of insects are reared together in a laboratory cage. They are supplied with two different types of food each day. Each individual of species 1 consumes 3 units of food \(A\) and 5 units of food \(B\), each individual of species 2 consumes 2 units of food \(A\) and 3 units of food \(B\), and individual of species 3 consumes 1 unit of food \(A\) and 2 units of food \(B\). Each day, 500 units of food \(A\) and 900 units of food \(B\) are supplied. How many individuals of each species can be reared together? Is there more than one solution? What happens if we add 550 units of a third type of food, called \(C\), and each individual of species 1 consumes 2 units of food \(C\), each individual of species 2 consumes 4 units of food \(C\), and each individual of species 3 consumes 1 unit of food \(C ?\)

$$ \begin{array}{r} A=\left[\begin{array}{rrr} 1 & 3 & -1 \\ 2 & 4 & 1 \\ 0 & -2 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 5 & -1 & 4 \\ 2 & 0 & 1 \\ 1 & -3 & -3 \end{array}\right], \\ \quad C=\left[\begin{array}{rrr} -2 & 0 & 4 \\ 1 & -3 & 1 \\ 0 & 0 & 2 \end{array}\right] \end{array} $$ $$ \text { Determine } D \text { so that } A+4 B=2(A+B)+D \text { . } $$
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