/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 Value of a \(2 \times 2\) determ... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 55

Value of a \(2 \times 2\) determinant If \(|a|\) is much greater than \(|b|,|c|\) and \(|d|,\) to which of \(a, b, c,\) and \(d\) is the value of the determinant $$f(a, b, c, d)=\left|\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right|$$ most sensitive? Give reasons for your answer.

Short Answer

Expert verified
The determinant is most sensitive to \(a\) because \(|a|\) is much larger than \(|b|\), \(|c|\), and \(|d|\).

Step by step solution

01

Introduction to Determinant Formula

The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) can be calculated using the formula: \(f(a, b, c, d) = ad - bc\). This formula involves multiplying and subtracting the matrix's elements.
02

Analyze Sensitivity to Each Variable

Assess how the determinant \(ad - bc\) changes with respect to small perturbations in \(a\), \(b\), \(c\), and \(d\), considering the condition \(|a| \gg |b|, |c|, |d|\). Since \(|a|\) is significantly larger than the other variables, changes in \(a\) are more pronounced in the product \(ad\) than in the product \(bc\).
03

Sensitivity to 'a'

The determinant is calculated with \(ad\) as the dominant term due to the condition \(|a| \gg |b|, |c|, |d|\). If \(a\) changes slightly, it affects \(ad\) significantly more than changes in \(b\), \(c\), or \(d\) affect \(bc\). Therefore, the determinant is most sensitive to changes in \(a\).

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Key Concepts

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

Matrix Algebra
Matrix algebra is a system of mathematical operations that can be performed on matrices, which are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. It forms a foundational element in many areas of mathematics and engineering, allowing us to perform complex calculations cohesively.

In the context of a 2x2 matrix, such as \[\begin{bmatrix} a & b \ c & d\end{bmatrix},\]there are specific operations, like addition, subtraction, multiplication, and finding determinants, that can be applied. Each operation follows particular rules.
  • **Addition** involves adding corresponding elements of matrices of the same dimension.
  • **Multiplication** blends the rows of one matrix with the columns of another to produce a new matrix.
  • **Determinants** provide a scalar value that simplifies understanding certain matrix properties, like invertibility.
Understanding matrix algebra is crucial as it provides the tools needed to solve linear equations systems, perform transformations, and much more. It serves as a bridge to more advanced topics such as vector spaces and eigenvalues.
Sensitivity Analysis
Sensitivity analysis is a method used to determine how different values of an independent variable affect a particular dependent variable under a given set of assumptions. In the realm of matrix algebra, it helps us understand the impact of small changes in the matrix's elements on its determinant.

Given the formula for the determinant of a 2x2 matrix, \(ad - bc\), sensitivity analysis will evaluate how changes in each component鈥擻(a\), \(b\), \(c\), and \(d\)鈥攁lter the result.
  • With a significantly large \(|a|\) compared to \(|b|\), \(|c|\), and \(|d|\), altering \(a\) will notably impact the term \(ad\) since it's the dominant multiplier.
  • Changes in \(b\), \(c\), and \(d\) are less impactful due to their relative size, making \(a\) the most sensitive element in this context.
Sensitivity analysis is vital in optimizing processes and ensuring stability in calculations involving matrices, as it helps pinpoint which variables most significantly affect the outcomes.
Determinant Formula
The determinant of a matrix is a unique number that provides valuable insights about the matrix's properties, such as whether the matrix has an inverse. For a 2x2 matrix, the determinant is computed using the formula:
\[f(a, b, c, d) = ad - bc\]This computation involves multiplying the diagonal elements (\(a\) and \(d\)) and subtracting the product of the off-diagonal elements (\(b\) and \(c\)).
  • The determinant being zero indicates that a matrix is singular, meaning it doesn't have an inverse and won't uniquely solve a system of linear equations.
  • A non-zero determinant suggests the matrix is invertible, ensuring it has unique solutions in applied contexts.
  • This formula highlights how varying elements like \(a\), though perhaps unintuitive initially, can disproportionately affect the matrix's overall identity due to the context or conditions applied, such as \(|a|\) being more significant.
Understanding the determinant formula allows for a richer comprehension of matrix behavior and its applications in various fields, such as physics, statistics, and economics.

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Most popular questions from this chapter

Submarine sinkings The data in Table 14.4 show the results of a historical study of German submarines sunk by the U.S. Navy during 16 consecutive months of World War II. The data given for each month are the number of reported sinkings and the number of actual sinkings. The number of submarines sunk was slightly greater than the Navy's reports implied. Find a least squares line for estimating the number of actual sinkings from the number of reported sinkings.

In Exercises \(51-56,\) find the limit of \(f\) as \((x, y) \rightarrow(0,0)\) or show that the limit does not exist. $$ f(x, y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}} $$

In Exercises \(25-30,\) find the linearization \(L(x, y)\) of the function at each point. $$ f(x, y)=e^{2 y-x} \text { at } \quad \text { a. }(0,0), \quad \text { b. }(1,2) $$

Change along a helix Find the derivative of \(f(x, y, z)=\) \(x^{2}+y^{2}+z^{2}\) in the direction of the unit tangent vector of the helix $$\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+t \mathbf{k}$$ at the points where \(t=-\pi / 4,0,\) and \(\pi / 4 .\) The function \(f\) gives the square of the distance from a point \(P(x, y, z)\) on the helix to the origin. The derivatives calculated here give the rates at which the square of the distance is changing with respect to \(t\) as \(P\) moves through the points where \(t=-\pi / 4,0,\) and \(\pi / 4\) .

Among all the points on the graph of \(z=10-x^{2}-y^{2}\) that lie above the plane \(x+2 y+3 z=0\) , find the point farthest from the plane.
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