/*! 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 4 Find all possible choices of \(c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 4

Find all possible choices of \(c\) that would make the following matrix singular: \\[ \left(\begin{array}{lll} 1 & 1 & 1 \\ 1 & 9 & c \\ 1 & c & 3 \end{array}\right) \\]

Short Answer

Expert verified
The possible choices of c that would make the matrix singular are \(\pm \sqrt{19}\).

Step by step solution

01

Write down the matrix and the formula for the determinant of a 3x3 matrix.

Let A be the given matrix: \\[ A=\left(\begin{array}{ccc} 1 & 1 & 1 \\\ 1 & 9 & c \\\ 1 & c & 3 \end{array}\right) \\] The determinant of a 3x3 matrix A is given by: \[ det(A) = a(ei鈭抐h)鈭抌(di鈭抐g)+c(dh鈭抏g) \]
02

Calculate the determinant of matrix A.

We will now substitute the values of the matrix elements into the formula for the determinant: \[ det(A) = 1(9\cdot3-c\cdot c)-1(1\cdot 3-1\cdot c)+1(1\cdot c-1\cdot 9) \] Simplifying the expression, we get: \[ det(A) = 27-c^2-c+1+c-9 \]
03

Set the determinant equal to zero and solve for c.

For the matrix A to be singular, its determinant must be equal to zero: \[ det(A) = 0 \] Substituting the expression for the determinant of A, we get: \[ 27-c^2-c+1+c-9 = 0 \] Simplify the equation: \[ c^2 - 19 = 0\] Now, find the values of c that satisfy this equation: \[ c = \pm \sqrt{19} \] So, the possible choices of c that would make the matrix singular are \(\pm \sqrt{19}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Prove that if a row or a column of an \(n \times n\) matrix \(A\) consists entirely of zeros, then \(\operatorname{det}(A)=0\)

Let \(A\) and \(B\) be \(2 \times 2\) matrices and let \\[ \begin{aligned} C &=\left(\begin{array}{ll} a_{11} & a_{12} \\ b_{21} & b_{22} \end{array}\right), \quad D=\left(\begin{array}{ll} b_{11} & b_{12} \\ a_{21} & a_{22} \end{array}\right) \\ E &=\left(\begin{array}{ll} 0 & \alpha \\ \beta & 0 \end{array}\right) \end{aligned} \\] (a) Show that \(\operatorname{det}(A+B)=\operatorname{det}(A)+\operatorname{det}(B)+\) \(\operatorname{det}(C)+\operatorname{det}(D)\) (b) Show that if \(B=E A\) then \(\operatorname{det}(A+B)=\) \(\operatorname{det}(A)+\operatorname{det}(B)\)

Let \(\mathbf{x}, \mathbf{y},\) and \(\mathbf{z}\) be vectors in \(\mathbb{R}^{3} .\) Show each of the following: (a) \(\mathbf{x} \times \mathbf{x}=\mathbf{0}\) (b) \(\mathbf{y} \times \mathbf{x}=-(\mathbf{x} \times \mathbf{y})\) (c) \(\mathbf{x} \times(\mathbf{y}+\mathbf{z})=(\mathbf{x} \times \mathbf{y})+(\mathbf{x} \times \mathbf{z})\) (d) \(\mathbf{z}^{T}(\mathbf{x} \times \mathbf{y})=\left|\begin{array}{lll}x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \\ z_{1} & z_{2} & z_{3}\end{array}\right|\)

Evaluate the following determinants by inspection: (a) \(\left|\begin{array}{ll}3 & 5 \\ 2 & 4\end{array}\right|\) (b) \(\left|\begin{array}{rrr}2 & 0 & 0 \\ 4 & 1 & 0 \\ 7 & 3 & -2\end{array}\right|\) (c) \(\left|\begin{array}{lll}3 & 0 & 0 \\ 2 & 1 & 1 \\ 1 & 2 & 2\end{array}\right|\) (d) \(\left|\begin{array}{llll}4 & 0 & 2 & 1 \\ 5 & 0 & 4 & 2 \\ 2 & 0 & 3 & 4 \\ 1 & 0 & 2 & 3\end{array}\right|\)

Suppose that a \(3 \times 3\) matrix \(A\) factors into a product \\[ \left[\begin{array}{ccc} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{array}\right]\left[\begin{array}{ccc} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{array}\right] \\] Determine the value of det(A).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.