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Chapter 7: Problem 26

Show that the scalar and matrices \(\alpha=-3, \quad A=\left(\begin{array}{rr}-2 & 4 \\ 4 & 0\end{array}\right), \quad\) and \(\quad B=\left(\begin{array}{rr}0 & -5 \\ -2 & 3\end{array}\right)\) satisfy the given identity. \((A B)^{T}=B^{T} A^{T}\)

Short Answer

Expert verified
The identity \((AB)^T = B^T A^T\) is verified as true.

Step by step solution

01

Compute AB

First, calculate the product of matrices \( A \) and \( B \):\[A = \begin{pmatrix} -2 & 4 \ 4 & 0 \end{pmatrix}, B = \begin{pmatrix} 0 & -5 \ -2 & 3 \end{pmatrix}\]Calculate each element of the resulting matrix:- Top left: \((-2)\times 0 + 4\times (-2) = 0 - 8 = -8\)- Top right: \((-2)\times (-5) + 4\times 3 = 10 + 12 = 22\)- Bottom left: \(4\times 0 + 0\times (-2) = 0\)- Bottom right: \(4\times (-5) + 0\times 3 = -20\)Thus, \( AB = \begin{pmatrix} -8 & 22 \ 0 & -20 \end{pmatrix} \).
02

Transpose of AB

Now, transpose the matrix \( AB \):If \( AB = \begin{pmatrix} -8 & 22 \ 0 & -20 \end{pmatrix} \), then the transpose is\[(AB)^T = \begin{pmatrix} -8 & 0 \ 22 & -20 \end{pmatrix}\].
03

Compute Transpose of B and A

Next, find the transpose of matrices \( B \) and \( A \):- \( B^T = \begin{pmatrix} 0 & -2 \ -5 & 3 \end{pmatrix} \)- \( A^T = \begin{pmatrix} -2 & 4 \ 4 & 0 \end{pmatrix} \) (Since \( A \) is already given, its transpose remains the same because it is symmetric)
04

Compute B^T A^T

Now calculate the product \( B^T A^T \):\[B^T = \begin{pmatrix} 0 & -2 \ -5 & 3 \end{pmatrix},A^T = \begin{pmatrix} -2 & 4 \ 4 & 0 \end{pmatrix}\]Calculate each element:- Top left: \(0 \times (-2) + (-2) \times 4 = 0 - 8 = -8\)- Top right: \(0 \times 4 + (-2) \times 0 = 0\)- Bottom left: \((-5) \times (-2) + 3 \times 4 = 10 + 12 = 22\)- Bottom right: \((-5) \times 4 + 3 \times 0 = -20\)Thus, \( B^T A^T = \begin{pmatrix} -8 & 0 \ 22 & -20 \end{pmatrix} \).
05

Verify Identity

Compare the results of \((AB)^T\) and \(B^T A^T\):\[(AB)^T = \begin{pmatrix} -8 & 0 \ 22 & -20 \end{pmatrix},B^T A^T = \begin{pmatrix} -8 & 0 \ 22 & -20 \end{pmatrix}\]The identity \((AB)^T = B^T A^T\) holds, as both matrices are equal.

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Key Concepts

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

Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra, essential for understanding various mathematical and real-life applications. When we multiply two matrices, we follow a specific, step-by-step process. Suppose we have matrix \(A\) with dimensions \(m \times n\) and matrix \(B\) with dimensions \(n \times p\). The resulting matrix, \(AB\), will have dimensions \(m \times p\).
  • To find an element in the resulting matrix, we multiply corresponding elements of the rows of \(A\) and the columns of \(B\), and then sum these products.
  • The number of columns in the first matrix (\(A\)) must be equal to the number of rows in the second matrix (\(B\)) for the multiplication to be valid.
  • Order matters in matrix multiplication, as \(AB\) is not necessarily equal to \(BA\).
Matrix multiplication is not only useful in pure math, but also in computer graphics, statistics, and many scientific computations. Understanding this operation opens the door to deeper exploration of matrix theory and its applications.
Transpose of a Matrix
The transpose of a matrix is an operation that flips a matrix over its diagonal, effectively switching the matrix's row and column indices. For a matrix \(M\) with dimensions \(m \times n\), its transpose, denoted \(M^T\), has dimensions \(n \times m\).
  • Each element at position \((i, j)\) in \(M\), becomes the element at position \((j, i)\) in \(M^T\).
  • Taking the transpose is particularly useful in solving problems involving symmetric matrices, where \(M = M^T\).
  • Transposing a transpose returns the original matrix: \((M^T)^T = M\).
The transpose operation is incredibly useful in many fields, including data science, where it helps in preparing and cleaning data for algorithms that expect a certain format. Transposing matrices is also pivotal when working with matrix identities, as seen in the exercise you are working with.
Matrix Identity Verification
Matrix identity verification involves proving relationships or properties among matrices through algebraic manipulations and operations. A common matrix identity is \((AB)^T = B^T A^T\). This states that the transpose of the product of two matrices is equal to the product of their transposes in reverse order.
  • To verify this identity, you calculate \(AB\), transpose it to get \((AB)^T\), and then compute \(B^T A^T\).
  • If \((AB)^T = B^T A^T\), the identity holds.
  • This property is crucial in many mathematical proofs and applications, such as in theoretical computations and transformations.
Verifying matrix identities helps in understanding more complex concepts and solving higher-level problems. It's also important for computational applications like simulations and modeling, where accuracy is crucial. Getting comfortable with these types of verifications allows you to tackle more challenging algebraic problems with confidence.

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Most popular questions from this chapter

Find the values of \(x\) for which the indicated matrix has a nontrivial nullspace. \(\left(\begin{array}{rr}2 & x \\ 3 & -2\end{array}\right)\)

Without actually solving, which systems have unique solutions? Explain. \(x_{1}+2 x_{2}=4\) \(2 x_{1}+4 x_{2}=8\)

Find all solutions of the homogeneous system \(A \mathbf{x}=\mathbf{0}\) for the given coefficient matrix. Does the system have solutions other than the zero vector? Use Proposition \(6.6\) to determine whether the matrix \(A\) is singular or nonsingular. \(A=\left(\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0\end{array}\right)\)

Compute the determinant of the matrix. In each case, decide if there is a nonzero vector in the nullspace. \(\left(\begin{array}{rrr}-1 & -9 & 10 \\ -7 & -19 & 26 \\ -2 & -10 & 12\end{array}\right)\)

Give a geometric description of the span of the given vectors in the given space. \(\mathbf{v}_{1}=(1,-2)^{T}, \mathbf{v}_{2}=(2,-4)^{T}\), in \(\mathbf{R}^{2}\)
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