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Chapter 3: Problem 18

\(\begin{aligned}&\text { Find (a) }|\boldsymbol{A}|,(\mathbf{b})|\boldsymbol{B}|,(\mathbf{c})\boldsymbol{A}+\boldsymbol{B}, \text { and }(\mathbf{d})|\boldsymbol{A}+\boldsymbol{B}| . \text { Then }\\\&\text { verify that}|\boldsymbol{A}|+|\boldsymbol{B}|\neq|\boldsymbol{A}+\boldsymbol{B}|\end{aligned}\). $$A=\left[\begin{array}{rrr}0 & 1 & 2 \\\1 & -1 & 0 \\\2 & 1 & 1\end{array}\right], \quad B=\left[\begin{array}{rrr}0 & 1 & -1 \\\2 & 1 & 1 \\\0 & 1 & 1\end{array}\right]$$

Short Answer

Expert verified
After performing calculations, we find the norm of matrix A (|A|), the norm of matrix B (|B|), the sum of the matrices A + B, and the norm of the resultant matrix |A + B|. Through comparison, it's evident that |A| + |B| is not equal to |A + B|, thus verifying the inequality.

Step by step solution

01

Compute Norm of Matrix A

The norm or determinant of a 3x3 matrix can be calculated using the formula: \[ \text {det}(A)=a(ei鈭抐h)鈭抌(di鈭抐g)+c(dh鈭抏g) \] By applying values from Matrix A into the formula, the determinant of Matrix A, |A|, can be computed.
02

Compute Norm of Matrix B

Repeat the same process with Matrix B to find |B|. Input the values from Matrix B into the determinant formula.
03

Perform Matrix Addition

Add Matrix A and Matrix B to produce a new matrix (A + B). This is done by adding the corresponding elements in Matrix A and B.
04

Compute Norm of the Resultant Matrix

Find the norm of the resulting matrix (A + B) by applying its values into the determinant formula. This will give us |A + B|.
05

Comparison of Norms

Compare if |A| + |B| is equal to |A + B|. If they are the same, then the verification is complete. If they are different, then the evidence has confirmed that |A| + |B| is not equal to |A + B|.

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Key Concepts

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

Matrix Operations
Understanding matrix operations is crucial in solving complex mathematical problems. Matrices are arrays of numbers and form the building blocks of linear algebra.
Matrix operations refer to manipulations involving matrices such as addition, subtraction, and multiplication. Each
operation has its own set of rules. For instance, to add matrices, both must have the same dimensions. When multiplying, the number of columns in the first matrix must match the number of rows in the second.
  • Identifying Matrices: Examine each matrix to understand its dimensions and values. This step ensures compatibility for further operations.
  • Transposition: Transform the rows into columns and columns into rows. It鈥檚 a basic but vital operation that precedes more complex processes.
Matrix operations make it easier to handle and solve linear equations, perform transformations, and even compute determinants.
Matrix Addition
Matrix addition is a simple, yet important operation. It involves adding two matrices element-wise.
This means summing corresponding elements from each matrix.
The process only works if both matrices are the same size. To perform matrix addition:
  • Check Dimensions: Both matrices must have the same number of rows and columns.
  • Add Corresponding Elements: Create a new matrix by adding elements in the same position from both matrices. For example, if A[1,1] is 5 and B[1,1] is 3, then the resulting matrix at that position is 5 + 3 = 8.
  • Result Matrix: This newly created matrix represents the sum of matrices A and B.
Matrix addition is used frequently in various applications including graphics, system simulations, and in setting up coupled equations.
Verification of Inequalities
Verification of inequalities in linear algebra often involves checking relationships between matrices and their determinants.
An important property of matrices is that the determinant, which is a special number calculated from its elements,
is not always preserved under certain operations like addition.In this exercise, the primary goal is to verify the inequality:
  • Calculating Determinants: First, calculate the determinants of individual matrices |A| and |B| using the determinant formula.
  • Addition of Determinants: Add these individual determinants |A| + |B|.
  • Resultant Determinant: Compute the determinant of the resulting matrix from the matrix addition |A + B|.
  • Verification: Compare the sum of the individual determinants with the resultant determinant. The inequality |A| + |B| \( eq \) |A + B| highlights that these calculated determinants are not equal, proving that determinant properties are intricate and can differ from straightforward arithmetic.
Verifying these inequalities aids in understanding more complex algebraic structures and supports breakthroughs in fields like computer graphics and engineering applications.

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Most popular questions from this chapter

Find the volume of the tetrahedron with the given vertices. $$(5,4,-3),(4,-6,-4),(-6,-6,-5),(0,0,10)$$

Find the volume of the tetrahedron with the given vertices. $$(3,-1,1),(4,-4,4),(1,1,1),(0,0,1)$$

Guided Proof Prove Property 3 of Theorem 3.3 When \(B\) is obtained from \(A\) by multiplying a row of \(A\) by a nonzero constant \(c, \operatorname{det}(B)=c \operatorname{det}(A)\) Getting Started: To prove that the determinant of \(B\) is cqual to \(c\) times the determinant of \(A,\) you need to show that the determinant of \(B\) is equal to \(c\) times the cofactor expansion of the determinant of \(A\). (i) Begin by letting \(B\) be the matrix obtained by multiplying \(c\) times the \(i\) th row of \(A\) (ii) Find the determinant of \(B\) by expanding in this ith row. (iii) Factor out the common factor \(c .\) (iv) Show that the result is \(c\) times the determinant of \(A\)

Use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. $$\begin{aligned}x_{1}-x_{2}+x_{3} &=4 \\\2 x_{1}-x_{2}+x_{3} &=6 \\\3 x_{1}-2 x_{2}+2 x_{3} &=0\end{aligned}$$

Prove that the determinant of an invertible matrix \(A\) is equal to \(\pm 1\) when all of the entries of \(A\) and \(A^{-1}\) are integers. Getting Started: Denote det( \(A\) ) as \(x\) and \(\operatorname{det}\left(A^{-1}\right)\) as \(y\) Note that \(x\) and \(y\) are real numbers. To prove that \(\operatorname{det}(A)\) is equal to \(\pm 1,\) you must show that both \(x\) and \(y\) are integers such that their product \(x y\) is equal to 1 (i) Use the property for the determinant of a matrix product to show that \(x y=1\) (ii) Use the definition of a determinant and the fact that the entries of \(A\) and \(A^{-1}\) are integers to show that both \(x=\operatorname{det}(A)\) and \(y=\operatorname{det}\left(A^{-1}\right)\) are integers. (iii) Conclude that \(x=\operatorname{det}(A)\) must be either 1 or \(-1\) because these are the only integer solutions to the equation \(x y=1\)
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